cardinal numbers and the continuum hypothesis · finite sizes infinite sizes cardinal arithmetic...
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![Page 1: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/1.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Cardinal Numbers and the ContinuumHypothesis
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 2: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/2.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Introduction
1. We want a standard “size” for each set, just like thenumber of elements (which is a natural number) is thestandard size for finite sets.
2. Ordinal numbers will not quite work because differentordinal numbers can have the same size.
3. Plus, once we are past that, we want to do arithmetic.4. To start, consider the arithmetic of finite set sizes.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 3: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/3.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Introduction1. We want a standard “size” for each set, just like the
number of elements (which is a natural number) is thestandard size for finite sets.
2. Ordinal numbers will not quite work because differentordinal numbers can have the same size.
3. Plus, once we are past that, we want to do arithmetic.4. To start, consider the arithmetic of finite set sizes.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 4: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/4.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Introduction1. We want a standard “size” for each set, just like the
number of elements (which is a natural number) is thestandard size for finite sets.
2. Ordinal numbers will not quite work because differentordinal numbers can have the same size.
3. Plus, once we are past that, we want to do arithmetic.4. To start, consider the arithmetic of finite set sizes.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 5: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/5.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Introduction1. We want a standard “size” for each set, just like the
number of elements (which is a natural number) is thestandard size for finite sets.
2. Ordinal numbers will not quite work because differentordinal numbers can have the same size.
3. Plus, once we are past that, we want to do arithmetic.
4. To start, consider the arithmetic of finite set sizes.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 6: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/6.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Introduction1. We want a standard “size” for each set, just like the
number of elements (which is a natural number) is thestandard size for finite sets.
2. Ordinal numbers will not quite work because differentordinal numbers can have the same size.
3. Plus, once we are past that, we want to do arithmetic.4. To start, consider the arithmetic of finite set sizes.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 7: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/7.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem.
Let A,B and C be finite sets. Then the followinghold.
1. If A∩B = /0, then |A∪B|= |A|+ |B|.2. |A∪B|= |A|+ |B|− |A∩B|.3.|A∪B∪C|= |A|+|B|+|C|−|B∩C|−|A∩B|−|A∩C|+|A∩B∩C|.
4. |A×B|= |A| · |B|.5. With AB denoting the set of all functions from B to A, we
have∣∣AB∣∣ = |A||B|.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 8: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/8.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let A,B and C be finite sets.
Then the followinghold.
1. If A∩B = /0, then |A∪B|= |A|+ |B|.2. |A∪B|= |A|+ |B|− |A∩B|.3.|A∪B∪C|= |A|+|B|+|C|−|B∩C|−|A∩B|−|A∩C|+|A∩B∩C|.
4. |A×B|= |A| · |B|.5. With AB denoting the set of all functions from B to A, we
have∣∣AB∣∣ = |A||B|.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 9: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/9.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let A,B and C be finite sets. Then the followinghold.
1. If A∩B = /0, then |A∪B|= |A|+ |B|.2. |A∪B|= |A|+ |B|− |A∩B|.3.|A∪B∪C|= |A|+|B|+|C|−|B∩C|−|A∩B|−|A∩C|+|A∩B∩C|.
4. |A×B|= |A| · |B|.5. With AB denoting the set of all functions from B to A, we
have∣∣AB∣∣ = |A||B|.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 10: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/10.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let A,B and C be finite sets. Then the followinghold.
1. If A∩B = /0, then |A∪B|= |A|+ |B|.
2. |A∪B|= |A|+ |B|− |A∩B|.3.|A∪B∪C|= |A|+|B|+|C|−|B∩C|−|A∩B|−|A∩C|+|A∩B∩C|.
4. |A×B|= |A| · |B|.5. With AB denoting the set of all functions from B to A, we
have∣∣AB∣∣ = |A||B|.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 11: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/11.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let A,B and C be finite sets. Then the followinghold.
1. If A∩B = /0, then |A∪B|= |A|+ |B|.2. |A∪B|= |A|+ |B|− |A∩B|.
3.|A∪B∪C|= |A|+|B|+|C|−|B∩C|−|A∩B|−|A∩C|+|A∩B∩C|.
4. |A×B|= |A| · |B|.5. With AB denoting the set of all functions from B to A, we
have∣∣AB∣∣ = |A||B|.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 12: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/12.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let A,B and C be finite sets. Then the followinghold.
1. If A∩B = /0, then |A∪B|= |A|+ |B|.2. |A∪B|= |A|+ |B|− |A∩B|.3.|A∪B∪C|= |A|+|B|+|C|−|B∩C|−|A∩B|−|A∩C|+|A∩B∩C|.
4. |A×B|= |A| · |B|.5. With AB denoting the set of all functions from B to A, we
have∣∣AB∣∣ = |A||B|.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 13: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/13.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let A,B and C be finite sets. Then the followinghold.
1. If A∩B = /0, then |A∪B|= |A|+ |B|.2. |A∪B|= |A|+ |B|− |A∩B|.3.|A∪B∪C|= |A|+|B|+|C|−|B∩C|−|A∩B|−|A∩C|+|A∩B∩C|.
4. |A×B|= |A| · |B|.
5. With AB denoting the set of all functions from B to A, wehave
∣∣AB∣∣ = |A||B|.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 14: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/14.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let A,B and C be finite sets. Then the followinghold.
1. If A∩B = /0, then |A∪B|= |A|+ |B|.2. |A∪B|= |A|+ |B|− |A∩B|.3.|A∪B∪C|= |A|+|B|+|C|−|B∩C|−|A∩B|−|A∩C|+|A∩B∩C|.
4. |A×B|= |A| · |B|.5. With AB denoting the set of all functions from B to A, we
have∣∣AB∣∣ = |A||B|.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 15: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/15.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (parts 2 and 3 only).
|A∪B|+ |A∩B| =∣∣A∪ (B\A)
∣∣+ |A∩B|= |A|+ |B\A|+ |A∩B|= |A|+
∣∣B\ (A∩B)∣∣+ |A∩B|
= |A|+ |B|
|A∪B∪C|=
∣∣A∪ (B∪C)∣∣
= |A|+ |B∪C|−∣∣A∩ (B∪C)
∣∣= |A|+ |B|+ |C|− |B∩C|−
∣∣(A∩B)∪ (A∩C)∣∣
= |A|+|B|+|C|−|B∩C|−[|A∩B|+|A∩C|−
∣∣(A∩B)∩(A∩C)∣∣]
= |A|+ |B|+ |C|− |B∩C|− |A∩B|− |A∩C|+ |A∩B∩C|
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 16: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/16.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (parts 2 and 3 only).|A∪B|+ |A∩B|
=∣∣A∪ (B\A)
∣∣+ |A∩B|= |A|+ |B\A|+ |A∩B|= |A|+
∣∣B\ (A∩B)∣∣+ |A∩B|
= |A|+ |B|
|A∪B∪C|=
∣∣A∪ (B∪C)∣∣
= |A|+ |B∪C|−∣∣A∩ (B∪C)
∣∣= |A|+ |B|+ |C|− |B∩C|−
∣∣(A∩B)∪ (A∩C)∣∣
= |A|+|B|+|C|−|B∩C|−[|A∩B|+|A∩C|−
∣∣(A∩B)∩(A∩C)∣∣]
= |A|+ |B|+ |C|− |B∩C|− |A∩B|− |A∩C|+ |A∩B∩C|
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 17: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/17.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (parts 2 and 3 only).|A∪B|+ |A∩B| =
∣∣A∪ (B\A)∣∣+ |A∩B|
= |A|+ |B\A|+ |A∩B|= |A|+
∣∣B\ (A∩B)∣∣+ |A∩B|
= |A|+ |B|
|A∪B∪C|=
∣∣A∪ (B∪C)∣∣
= |A|+ |B∪C|−∣∣A∩ (B∪C)
∣∣= |A|+ |B|+ |C|− |B∩C|−
∣∣(A∩B)∪ (A∩C)∣∣
= |A|+|B|+|C|−|B∩C|−[|A∩B|+|A∩C|−
∣∣(A∩B)∩(A∩C)∣∣]
= |A|+ |B|+ |C|− |B∩C|− |A∩B|− |A∩C|+ |A∩B∩C|
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 18: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/18.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (parts 2 and 3 only).|A∪B|+ |A∩B| =
∣∣A∪ (B\A)∣∣+ |A∩B|
= |A|+ |B\A|+ |A∩B|
= |A|+∣∣B\ (A∩B)
∣∣+ |A∩B|= |A|+ |B|
|A∪B∪C|=
∣∣A∪ (B∪C)∣∣
= |A|+ |B∪C|−∣∣A∩ (B∪C)
∣∣= |A|+ |B|+ |C|− |B∩C|−
∣∣(A∩B)∪ (A∩C)∣∣
= |A|+|B|+|C|−|B∩C|−[|A∩B|+|A∩C|−
∣∣(A∩B)∩(A∩C)∣∣]
= |A|+ |B|+ |C|− |B∩C|− |A∩B|− |A∩C|+ |A∩B∩C|
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 19: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/19.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (parts 2 and 3 only).|A∪B|+ |A∩B| =
∣∣A∪ (B\A)∣∣+ |A∩B|
= |A|+ |B\A|+ |A∩B|= |A|+
∣∣B\ (A∩B)∣∣+ |A∩B|
= |A|+ |B|
|A∪B∪C|=
∣∣A∪ (B∪C)∣∣
= |A|+ |B∪C|−∣∣A∩ (B∪C)
∣∣= |A|+ |B|+ |C|− |B∩C|−
∣∣(A∩B)∪ (A∩C)∣∣
= |A|+|B|+|C|−|B∩C|−[|A∩B|+|A∩C|−
∣∣(A∩B)∩(A∩C)∣∣]
= |A|+ |B|+ |C|− |B∩C|− |A∩B|− |A∩C|+ |A∩B∩C|
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 20: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/20.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (parts 2 and 3 only).|A∪B|+ |A∩B| =
∣∣A∪ (B\A)∣∣+ |A∩B|
= |A|+ |B\A|+ |A∩B|= |A|+
∣∣B\ (A∩B)∣∣+ |A∩B|
= |A|+ |B|
|A∪B∪C|=
∣∣A∪ (B∪C)∣∣
= |A|+ |B∪C|−∣∣A∩ (B∪C)
∣∣= |A|+ |B|+ |C|− |B∩C|−
∣∣(A∩B)∪ (A∩C)∣∣
= |A|+|B|+|C|−|B∩C|−[|A∩B|+|A∩C|−
∣∣(A∩B)∩(A∩C)∣∣]
= |A|+ |B|+ |C|− |B∩C|− |A∩B|− |A∩C|+ |A∩B∩C|
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 21: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/21.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (parts 2 and 3 only).|A∪B|+ |A∩B| =
∣∣A∪ (B\A)∣∣+ |A∩B|
= |A|+ |B\A|+ |A∩B|= |A|+
∣∣B\ (A∩B)∣∣+ |A∩B|
= |A|+ |B|
|A∪B∪C|
=∣∣A∪ (B∪C)
∣∣= |A|+ |B∪C|−
∣∣A∩ (B∪C)∣∣
= |A|+ |B|+ |C|− |B∩C|−∣∣(A∩B)∪ (A∩C)
∣∣= |A|+|B|+|C|−|B∩C|−
[|A∩B|+|A∩C|−
∣∣(A∩B)∩(A∩C)∣∣]
= |A|+ |B|+ |C|− |B∩C|− |A∩B|− |A∩C|+ |A∩B∩C|
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 22: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/22.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (parts 2 and 3 only).|A∪B|+ |A∩B| =
∣∣A∪ (B\A)∣∣+ |A∩B|
= |A|+ |B\A|+ |A∩B|= |A|+
∣∣B\ (A∩B)∣∣+ |A∩B|
= |A|+ |B|
|A∪B∪C|=
∣∣A∪ (B∪C)∣∣
= |A|+ |B∪C|−∣∣A∩ (B∪C)
∣∣= |A|+ |B|+ |C|− |B∩C|−
∣∣(A∩B)∪ (A∩C)∣∣
= |A|+|B|+|C|−|B∩C|−[|A∩B|+|A∩C|−
∣∣(A∩B)∩(A∩C)∣∣]
= |A|+ |B|+ |C|− |B∩C|− |A∩B|− |A∩C|+ |A∩B∩C|
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 23: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/23.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (parts 2 and 3 only).|A∪B|+ |A∩B| =
∣∣A∪ (B\A)∣∣+ |A∩B|
= |A|+ |B\A|+ |A∩B|= |A|+
∣∣B\ (A∩B)∣∣+ |A∩B|
= |A|+ |B|
|A∪B∪C|=
∣∣A∪ (B∪C)∣∣
= |A|+ |B∪C|−∣∣A∩ (B∪C)
∣∣
= |A|+ |B|+ |C|− |B∩C|−∣∣(A∩B)∪ (A∩C)
∣∣= |A|+|B|+|C|−|B∩C|−
[|A∩B|+|A∩C|−
∣∣(A∩B)∩(A∩C)∣∣]
= |A|+ |B|+ |C|− |B∩C|− |A∩B|− |A∩C|+ |A∩B∩C|
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 24: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/24.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (parts 2 and 3 only).|A∪B|+ |A∩B| =
∣∣A∪ (B\A)∣∣+ |A∩B|
= |A|+ |B\A|+ |A∩B|= |A|+
∣∣B\ (A∩B)∣∣+ |A∩B|
= |A|+ |B|
|A∪B∪C|=
∣∣A∪ (B∪C)∣∣
= |A|+ |B∪C|−∣∣A∩ (B∪C)
∣∣= |A|+ |B|+ |C|− |B∩C|−
∣∣(A∩B)∪ (A∩C)∣∣
= |A|+|B|+|C|−|B∩C|−[|A∩B|+|A∩C|−
∣∣(A∩B)∩(A∩C)∣∣]
= |A|+ |B|+ |C|− |B∩C|− |A∩B|− |A∩C|+ |A∩B∩C|
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 25: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/25.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (parts 2 and 3 only).|A∪B|+ |A∩B| =
∣∣A∪ (B\A)∣∣+ |A∩B|
= |A|+ |B\A|+ |A∩B|= |A|+
∣∣B\ (A∩B)∣∣+ |A∩B|
= |A|+ |B|
|A∪B∪C|=
∣∣A∪ (B∪C)∣∣
= |A|+ |B∪C|−∣∣A∩ (B∪C)
∣∣= |A|+ |B|+ |C|− |B∩C|−
∣∣(A∩B)∪ (A∩C)∣∣
= |A|+|B|+|C|−|B∩C|−[|A∩B|+|A∩C|−
∣∣(A∩B)∩(A∩C)∣∣]
= |A|+ |B|+ |C|− |B∩C|− |A∩B|− |A∩C|+ |A∩B∩C|
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 26: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/26.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (parts 2 and 3 only).|A∪B|+ |A∩B| =
∣∣A∪ (B\A)∣∣+ |A∩B|
= |A|+ |B\A|+ |A∩B|= |A|+
∣∣B\ (A∩B)∣∣+ |A∩B|
= |A|+ |B|
|A∪B∪C|=
∣∣A∪ (B∪C)∣∣
= |A|+ |B∪C|−∣∣A∩ (B∪C)
∣∣= |A|+ |B|+ |C|− |B∩C|−
∣∣(A∩B)∪ (A∩C)∣∣
= |A|+|B|+|C|−|B∩C|−[|A∩B|+|A∩C|−
∣∣(A∩B)∩(A∩C)∣∣]
= |A|+ |B|+ |C|− |B∩C|− |A∩B|− |A∩C|+ |A∩B∩C|
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 27: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/27.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (parts 2 and 3 only).|A∪B|+ |A∩B| =
∣∣A∪ (B\A)∣∣+ |A∩B|
= |A|+ |B\A|+ |A∩B|= |A|+
∣∣B\ (A∩B)∣∣+ |A∩B|
= |A|+ |B|
|A∪B∪C|=
∣∣A∪ (B∪C)∣∣
= |A|+ |B∪C|−∣∣A∩ (B∪C)
∣∣= |A|+ |B|+ |C|− |B∩C|−
∣∣(A∩B)∪ (A∩C)∣∣
= |A|+|B|+|C|−|B∩C|−[|A∩B|+|A∩C|−
∣∣(A∩B)∩(A∩C)∣∣]
= |A|+ |B|+ |C|− |B∩C|− |A∩B|− |A∩C|+ |A∩B∩C|
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 28: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/28.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Example.
In the restaurant “Zum Adler” 41 people are dining.25 people have ordered bone marrow dumpling soup as anappetizer. 32 people have ordered blood sausage as the maincourse. 18 people have ordered black forest cake for desert. 12people will have bone marrow dumpling soup and bloodsausage. 9 people will have bone marrow dumpling soup andblack forest cake. 15 people will have blood sausage and blackforest cake. How many people will have all three dishes?M :=set of all people who will have bone marrow dumplingsoup. S :=set of all people who will have blood sausage.C :=set of all people who will have black forest cake.
|M∩S∩C|= |M∪S∪C|−|M|−|S|−|C|+|M∩S|+|M∩C|+|S∩C|= 41−25−32−18+12+9+15 = 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 29: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/29.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Example. In the restaurant “Zum Adler” 41 people are dining.
25 people have ordered bone marrow dumpling soup as anappetizer. 32 people have ordered blood sausage as the maincourse. 18 people have ordered black forest cake for desert. 12people will have bone marrow dumpling soup and bloodsausage. 9 people will have bone marrow dumpling soup andblack forest cake. 15 people will have blood sausage and blackforest cake. How many people will have all three dishes?M :=set of all people who will have bone marrow dumplingsoup. S :=set of all people who will have blood sausage.C :=set of all people who will have black forest cake.
|M∩S∩C|= |M∪S∪C|−|M|−|S|−|C|+|M∩S|+|M∩C|+|S∩C|= 41−25−32−18+12+9+15 = 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 30: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/30.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Example. In the restaurant “Zum Adler” 41 people are dining.25 people have ordered bone marrow dumpling soup as anappetizer.
32 people have ordered blood sausage as the maincourse. 18 people have ordered black forest cake for desert. 12people will have bone marrow dumpling soup and bloodsausage. 9 people will have bone marrow dumpling soup andblack forest cake. 15 people will have blood sausage and blackforest cake. How many people will have all three dishes?M :=set of all people who will have bone marrow dumplingsoup. S :=set of all people who will have blood sausage.C :=set of all people who will have black forest cake.
|M∩S∩C|= |M∪S∪C|−|M|−|S|−|C|+|M∩S|+|M∩C|+|S∩C|= 41−25−32−18+12+9+15 = 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 31: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/31.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Example. In the restaurant “Zum Adler” 41 people are dining.25 people have ordered bone marrow dumpling soup as anappetizer. 32 people have ordered blood sausage as the maincourse.
18 people have ordered black forest cake for desert. 12people will have bone marrow dumpling soup and bloodsausage. 9 people will have bone marrow dumpling soup andblack forest cake. 15 people will have blood sausage and blackforest cake. How many people will have all three dishes?M :=set of all people who will have bone marrow dumplingsoup. S :=set of all people who will have blood sausage.C :=set of all people who will have black forest cake.
|M∩S∩C|= |M∪S∪C|−|M|−|S|−|C|+|M∩S|+|M∩C|+|S∩C|= 41−25−32−18+12+9+15 = 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 32: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/32.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Example. In the restaurant “Zum Adler” 41 people are dining.25 people have ordered bone marrow dumpling soup as anappetizer. 32 people have ordered blood sausage as the maincourse. 18 people have ordered black forest cake for desert.
12people will have bone marrow dumpling soup and bloodsausage. 9 people will have bone marrow dumpling soup andblack forest cake. 15 people will have blood sausage and blackforest cake. How many people will have all three dishes?M :=set of all people who will have bone marrow dumplingsoup. S :=set of all people who will have blood sausage.C :=set of all people who will have black forest cake.
|M∩S∩C|= |M∪S∪C|−|M|−|S|−|C|+|M∩S|+|M∩C|+|S∩C|= 41−25−32−18+12+9+15 = 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 33: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/33.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Example. In the restaurant “Zum Adler” 41 people are dining.25 people have ordered bone marrow dumpling soup as anappetizer. 32 people have ordered blood sausage as the maincourse. 18 people have ordered black forest cake for desert. 12people will have bone marrow dumpling soup and bloodsausage.
9 people will have bone marrow dumpling soup andblack forest cake. 15 people will have blood sausage and blackforest cake. How many people will have all three dishes?M :=set of all people who will have bone marrow dumplingsoup. S :=set of all people who will have blood sausage.C :=set of all people who will have black forest cake.
|M∩S∩C|= |M∪S∪C|−|M|−|S|−|C|+|M∩S|+|M∩C|+|S∩C|= 41−25−32−18+12+9+15 = 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 34: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/34.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Example. In the restaurant “Zum Adler” 41 people are dining.25 people have ordered bone marrow dumpling soup as anappetizer. 32 people have ordered blood sausage as the maincourse. 18 people have ordered black forest cake for desert. 12people will have bone marrow dumpling soup and bloodsausage. 9 people will have bone marrow dumpling soup andblack forest cake.
15 people will have blood sausage and blackforest cake. How many people will have all three dishes?M :=set of all people who will have bone marrow dumplingsoup. S :=set of all people who will have blood sausage.C :=set of all people who will have black forest cake.
|M∩S∩C|= |M∪S∪C|−|M|−|S|−|C|+|M∩S|+|M∩C|+|S∩C|= 41−25−32−18+12+9+15 = 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 35: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/35.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Example. In the restaurant “Zum Adler” 41 people are dining.25 people have ordered bone marrow dumpling soup as anappetizer. 32 people have ordered blood sausage as the maincourse. 18 people have ordered black forest cake for desert. 12people will have bone marrow dumpling soup and bloodsausage. 9 people will have bone marrow dumpling soup andblack forest cake. 15 people will have blood sausage and blackforest cake.
How many people will have all three dishes?M :=set of all people who will have bone marrow dumplingsoup. S :=set of all people who will have blood sausage.C :=set of all people who will have black forest cake.
|M∩S∩C|= |M∪S∪C|−|M|−|S|−|C|+|M∩S|+|M∩C|+|S∩C|= 41−25−32−18+12+9+15 = 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 36: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/36.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Example. In the restaurant “Zum Adler” 41 people are dining.25 people have ordered bone marrow dumpling soup as anappetizer. 32 people have ordered blood sausage as the maincourse. 18 people have ordered black forest cake for desert. 12people will have bone marrow dumpling soup and bloodsausage. 9 people will have bone marrow dumpling soup andblack forest cake. 15 people will have blood sausage and blackforest cake. How many people will have all three dishes?
M :=set of all people who will have bone marrow dumplingsoup. S :=set of all people who will have blood sausage.C :=set of all people who will have black forest cake.
|M∩S∩C|= |M∪S∪C|−|M|−|S|−|C|+|M∩S|+|M∩C|+|S∩C|= 41−25−32−18+12+9+15 = 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 37: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/37.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Example. In the restaurant “Zum Adler” 41 people are dining.25 people have ordered bone marrow dumpling soup as anappetizer. 32 people have ordered blood sausage as the maincourse. 18 people have ordered black forest cake for desert. 12people will have bone marrow dumpling soup and bloodsausage. 9 people will have bone marrow dumpling soup andblack forest cake. 15 people will have blood sausage and blackforest cake. How many people will have all three dishes?M :=set of all people who will have bone marrow dumplingsoup.
S :=set of all people who will have blood sausage.C :=set of all people who will have black forest cake.
|M∩S∩C|= |M∪S∪C|−|M|−|S|−|C|+|M∩S|+|M∩C|+|S∩C|= 41−25−32−18+12+9+15 = 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 38: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/38.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Example. In the restaurant “Zum Adler” 41 people are dining.25 people have ordered bone marrow dumpling soup as anappetizer. 32 people have ordered blood sausage as the maincourse. 18 people have ordered black forest cake for desert. 12people will have bone marrow dumpling soup and bloodsausage. 9 people will have bone marrow dumpling soup andblack forest cake. 15 people will have blood sausage and blackforest cake. How many people will have all three dishes?M :=set of all people who will have bone marrow dumplingsoup. S :=set of all people who will have blood sausage.
C :=set of all people who will have black forest cake.
|M∩S∩C|= |M∪S∪C|−|M|−|S|−|C|+|M∩S|+|M∩C|+|S∩C|= 41−25−32−18+12+9+15 = 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 39: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/39.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Example. In the restaurant “Zum Adler” 41 people are dining.25 people have ordered bone marrow dumpling soup as anappetizer. 32 people have ordered blood sausage as the maincourse. 18 people have ordered black forest cake for desert. 12people will have bone marrow dumpling soup and bloodsausage. 9 people will have bone marrow dumpling soup andblack forest cake. 15 people will have blood sausage and blackforest cake. How many people will have all three dishes?M :=set of all people who will have bone marrow dumplingsoup. S :=set of all people who will have blood sausage.C :=set of all people who will have black forest cake.
|M∩S∩C|= |M∪S∪C|−|M|−|S|−|C|+|M∩S|+|M∩C|+|S∩C|= 41−25−32−18+12+9+15 = 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 40: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/40.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Example. In the restaurant “Zum Adler” 41 people are dining.25 people have ordered bone marrow dumpling soup as anappetizer. 32 people have ordered blood sausage as the maincourse. 18 people have ordered black forest cake for desert. 12people will have bone marrow dumpling soup and bloodsausage. 9 people will have bone marrow dumpling soup andblack forest cake. 15 people will have blood sausage and blackforest cake. How many people will have all three dishes?M :=set of all people who will have bone marrow dumplingsoup. S :=set of all people who will have blood sausage.C :=set of all people who will have black forest cake.
|M∩S∩C|
= |M∪S∪C|−|M|−|S|−|C|+|M∩S|+|M∩C|+|S∩C|= 41−25−32−18+12+9+15 = 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 41: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/41.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Example. In the restaurant “Zum Adler” 41 people are dining.25 people have ordered bone marrow dumpling soup as anappetizer. 32 people have ordered blood sausage as the maincourse. 18 people have ordered black forest cake for desert. 12people will have bone marrow dumpling soup and bloodsausage. 9 people will have bone marrow dumpling soup andblack forest cake. 15 people will have blood sausage and blackforest cake. How many people will have all three dishes?M :=set of all people who will have bone marrow dumplingsoup. S :=set of all people who will have blood sausage.C :=set of all people who will have black forest cake.
|M∩S∩C|= |M∪S∪C|−|M|−|S|−|C|+|M∩S|+|M∩C|+|S∩C|
= 41−25−32−18+12+9+15 = 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 42: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/42.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Example. In the restaurant “Zum Adler” 41 people are dining.25 people have ordered bone marrow dumpling soup as anappetizer. 32 people have ordered blood sausage as the maincourse. 18 people have ordered black forest cake for desert. 12people will have bone marrow dumpling soup and bloodsausage. 9 people will have bone marrow dumpling soup andblack forest cake. 15 people will have blood sausage and blackforest cake. How many people will have all three dishes?M :=set of all people who will have bone marrow dumplingsoup. S :=set of all people who will have blood sausage.C :=set of all people who will have black forest cake.
|M∩S∩C|= |M∪S∪C|−|M|−|S|−|C|+|M∩S|+|M∩C|+|S∩C|= 41−25−32−18+12+9+15
= 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 43: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/43.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Example. In the restaurant “Zum Adler” 41 people are dining.25 people have ordered bone marrow dumpling soup as anappetizer. 32 people have ordered blood sausage as the maincourse. 18 people have ordered black forest cake for desert. 12people will have bone marrow dumpling soup and bloodsausage. 9 people will have bone marrow dumpling soup andblack forest cake. 15 people will have blood sausage and blackforest cake. How many people will have all three dishes?M :=set of all people who will have bone marrow dumplingsoup. S :=set of all people who will have blood sausage.C :=set of all people who will have black forest cake.
|M∩S∩C|= |M∪S∪C|−|M|−|S|−|C|+|M∩S|+|M∩C|+|S∩C|= 41−25−32−18+12+9+15 = 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 44: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/44.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Example. In the restaurant “Zum Adler” 41 people are dining.25 people have ordered bone marrow dumpling soup as anappetizer. 32 people have ordered blood sausage as the maincourse. 18 people have ordered black forest cake for desert. 12people will have bone marrow dumpling soup and bloodsausage. 9 people will have bone marrow dumpling soup andblack forest cake. 15 people will have blood sausage and blackforest cake. How many people will have all three dishes?M :=set of all people who will have bone marrow dumplingsoup. S :=set of all people who will have blood sausage.C :=set of all people who will have black forest cake.
|M∩S∩C|= |M∪S∪C|−|M|−|S|−|C|+|M∩S|+|M∩C|+|S∩C|= 41−25−32−18+12+9+15 = 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 45: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/45.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Definition.
A cardinal number is an ordinal number α so thatfor all ordinal numbers β that are equivalent to α we haveα ⊆ β .
Definition. For every infinite set S we define the cardinality |S|of S to be the unique cardinal number α that is equivalent to S.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 46: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/46.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Definition. A cardinal number is an ordinal number α so thatfor all ordinal numbers β that are equivalent to α we haveα ⊆ β .
Definition. For every infinite set S we define the cardinality |S|of S to be the unique cardinal number α that is equivalent to S.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 47: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/47.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Definition. A cardinal number is an ordinal number α so thatfor all ordinal numbers β that are equivalent to α we haveα ⊆ β .
Definition.
For every infinite set S we define the cardinality |S|of S to be the unique cardinal number α that is equivalent to S.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 48: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/48.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Definition. A cardinal number is an ordinal number α so thatfor all ordinal numbers β that are equivalent to α we haveα ⊆ β .
Definition. For every infinite set S we define the cardinality |S|of S to be the unique cardinal number α that is equivalent to S.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 49: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/49.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem.
Cantor-Schroder-Bernstein Theorem. Let A and Bbe sets so that there is an injective function f : A → B and aninjective function g : B → A. Then there is a bijective functionh : A → B.
Proof. Define F(X) := A\g[B\ f [X]
]for all X ⊆ A. Then
X ⊆ Y implies f [X]⊆ f [Y], B\ f [X]⊇ B\ f [Y],g[B\ f [X]
]⊇ g
[B\ f [Y]
],
F(X) = A\g[B\ f [X]
]⊆ A\g
[B\ f [Y]
]= F(Y).
Let C :=⋃{
H ∈P(A) : H ⊆ F(H)}
and let c ∈ C. Then thereis an H ∈P(A) with c ∈ H ⊆ F(H)⊆ F(C). Hence C ⊆ F(C).Then F(C)⊆ F
(F(C)
). By definition of C, F(C)⊆ C. Thus
C = F(C).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 50: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/50.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Cantor-Schroder-Bernstein Theorem.
Let A and Bbe sets so that there is an injective function f : A → B and aninjective function g : B → A. Then there is a bijective functionh : A → B.
Proof. Define F(X) := A\g[B\ f [X]
]for all X ⊆ A. Then
X ⊆ Y implies f [X]⊆ f [Y], B\ f [X]⊇ B\ f [Y],g[B\ f [X]
]⊇ g
[B\ f [Y]
],
F(X) = A\g[B\ f [X]
]⊆ A\g
[B\ f [Y]
]= F(Y).
Let C :=⋃{
H ∈P(A) : H ⊆ F(H)}
and let c ∈ C. Then thereis an H ∈P(A) with c ∈ H ⊆ F(H)⊆ F(C). Hence C ⊆ F(C).Then F(C)⊆ F
(F(C)
). By definition of C, F(C)⊆ C. Thus
C = F(C).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 51: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/51.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Cantor-Schroder-Bernstein Theorem. Let A and Bbe sets so that there is an injective function f : A → B and aninjective function g : B → A.
Then there is a bijective functionh : A → B.
Proof. Define F(X) := A\g[B\ f [X]
]for all X ⊆ A. Then
X ⊆ Y implies f [X]⊆ f [Y], B\ f [X]⊇ B\ f [Y],g[B\ f [X]
]⊇ g
[B\ f [Y]
],
F(X) = A\g[B\ f [X]
]⊆ A\g
[B\ f [Y]
]= F(Y).
Let C :=⋃{
H ∈P(A) : H ⊆ F(H)}
and let c ∈ C. Then thereis an H ∈P(A) with c ∈ H ⊆ F(H)⊆ F(C). Hence C ⊆ F(C).Then F(C)⊆ F
(F(C)
). By definition of C, F(C)⊆ C. Thus
C = F(C).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 52: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/52.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Cantor-Schroder-Bernstein Theorem. Let A and Bbe sets so that there is an injective function f : A → B and aninjective function g : B → A. Then there is a bijective functionh : A → B.
Proof. Define F(X) := A\g[B\ f [X]
]for all X ⊆ A. Then
X ⊆ Y implies f [X]⊆ f [Y], B\ f [X]⊇ B\ f [Y],g[B\ f [X]
]⊇ g
[B\ f [Y]
],
F(X) = A\g[B\ f [X]
]⊆ A\g
[B\ f [Y]
]= F(Y).
Let C :=⋃{
H ∈P(A) : H ⊆ F(H)}
and let c ∈ C. Then thereis an H ∈P(A) with c ∈ H ⊆ F(H)⊆ F(C). Hence C ⊆ F(C).Then F(C)⊆ F
(F(C)
). By definition of C, F(C)⊆ C. Thus
C = F(C).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 53: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/53.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Cantor-Schroder-Bernstein Theorem. Let A and Bbe sets so that there is an injective function f : A → B and aninjective function g : B → A. Then there is a bijective functionh : A → B.
Proof.
Define F(X) := A\g[B\ f [X]
]for all X ⊆ A. Then
X ⊆ Y implies f [X]⊆ f [Y], B\ f [X]⊇ B\ f [Y],g[B\ f [X]
]⊇ g
[B\ f [Y]
],
F(X) = A\g[B\ f [X]
]⊆ A\g
[B\ f [Y]
]= F(Y).
Let C :=⋃{
H ∈P(A) : H ⊆ F(H)}
and let c ∈ C. Then thereis an H ∈P(A) with c ∈ H ⊆ F(H)⊆ F(C). Hence C ⊆ F(C).Then F(C)⊆ F
(F(C)
). By definition of C, F(C)⊆ C. Thus
C = F(C).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 54: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/54.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Cantor-Schroder-Bernstein Theorem. Let A and Bbe sets so that there is an injective function f : A → B and aninjective function g : B → A. Then there is a bijective functionh : A → B.
Proof. Define F(X) := A\g[B\ f [X]
]for all X ⊆ A.
ThenX ⊆ Y implies f [X]⊆ f [Y], B\ f [X]⊇ B\ f [Y],g[B\ f [X]
]⊇ g
[B\ f [Y]
],
F(X) = A\g[B\ f [X]
]⊆ A\g
[B\ f [Y]
]= F(Y).
Let C :=⋃{
H ∈P(A) : H ⊆ F(H)}
and let c ∈ C. Then thereis an H ∈P(A) with c ∈ H ⊆ F(H)⊆ F(C). Hence C ⊆ F(C).Then F(C)⊆ F
(F(C)
). By definition of C, F(C)⊆ C. Thus
C = F(C).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 55: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/55.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Cantor-Schroder-Bernstein Theorem. Let A and Bbe sets so that there is an injective function f : A → B and aninjective function g : B → A. Then there is a bijective functionh : A → B.
Proof. Define F(X) := A\g[B\ f [X]
]for all X ⊆ A. Then
X ⊆ Y implies f [X]⊆ f [Y]
, B\ f [X]⊇ B\ f [Y],g[B\ f [X]
]⊇ g
[B\ f [Y]
],
F(X) = A\g[B\ f [X]
]⊆ A\g
[B\ f [Y]
]= F(Y).
Let C :=⋃{
H ∈P(A) : H ⊆ F(H)}
and let c ∈ C. Then thereis an H ∈P(A) with c ∈ H ⊆ F(H)⊆ F(C). Hence C ⊆ F(C).Then F(C)⊆ F
(F(C)
). By definition of C, F(C)⊆ C. Thus
C = F(C).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 56: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/56.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Cantor-Schroder-Bernstein Theorem. Let A and Bbe sets so that there is an injective function f : A → B and aninjective function g : B → A. Then there is a bijective functionh : A → B.
Proof. Define F(X) := A\g[B\ f [X]
]for all X ⊆ A. Then
X ⊆ Y implies f [X]⊆ f [Y], B\ f [X]⊇ B\ f [Y]
,g[B\ f [X]
]⊇ g
[B\ f [Y]
],
F(X) = A\g[B\ f [X]
]⊆ A\g
[B\ f [Y]
]= F(Y).
Let C :=⋃{
H ∈P(A) : H ⊆ F(H)}
and let c ∈ C. Then thereis an H ∈P(A) with c ∈ H ⊆ F(H)⊆ F(C). Hence C ⊆ F(C).Then F(C)⊆ F
(F(C)
). By definition of C, F(C)⊆ C. Thus
C = F(C).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 57: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/57.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Cantor-Schroder-Bernstein Theorem. Let A and Bbe sets so that there is an injective function f : A → B and aninjective function g : B → A. Then there is a bijective functionh : A → B.
Proof. Define F(X) := A\g[B\ f [X]
]for all X ⊆ A. Then
X ⊆ Y implies f [X]⊆ f [Y], B\ f [X]⊇ B\ f [Y],g[B\ f [X]
]⊇ g
[B\ f [Y]
]
,F(X) = A\g
[B\ f [X]
]⊆ A\g
[B\ f [Y]
]= F(Y).
Let C :=⋃{
H ∈P(A) : H ⊆ F(H)}
and let c ∈ C. Then thereis an H ∈P(A) with c ∈ H ⊆ F(H)⊆ F(C). Hence C ⊆ F(C).Then F(C)⊆ F
(F(C)
). By definition of C, F(C)⊆ C. Thus
C = F(C).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 58: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/58.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Cantor-Schroder-Bernstein Theorem. Let A and Bbe sets so that there is an injective function f : A → B and aninjective function g : B → A. Then there is a bijective functionh : A → B.
Proof. Define F(X) := A\g[B\ f [X]
]for all X ⊆ A. Then
X ⊆ Y implies f [X]⊆ f [Y], B\ f [X]⊇ B\ f [Y],g[B\ f [X]
]⊇ g
[B\ f [Y]
],
F(X)
= A\g[B\ f [X]
]⊆ A\g
[B\ f [Y]
]= F(Y).
Let C :=⋃{
H ∈P(A) : H ⊆ F(H)}
and let c ∈ C. Then thereis an H ∈P(A) with c ∈ H ⊆ F(H)⊆ F(C). Hence C ⊆ F(C).Then F(C)⊆ F
(F(C)
). By definition of C, F(C)⊆ C. Thus
C = F(C).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 59: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/59.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Cantor-Schroder-Bernstein Theorem. Let A and Bbe sets so that there is an injective function f : A → B and aninjective function g : B → A. Then there is a bijective functionh : A → B.
Proof. Define F(X) := A\g[B\ f [X]
]for all X ⊆ A. Then
X ⊆ Y implies f [X]⊆ f [Y], B\ f [X]⊇ B\ f [Y],g[B\ f [X]
]⊇ g
[B\ f [Y]
],
F(X) = A\g[B\ f [X]
]
⊆ A\g[B\ f [Y]
]= F(Y).
Let C :=⋃{
H ∈P(A) : H ⊆ F(H)}
and let c ∈ C. Then thereis an H ∈P(A) with c ∈ H ⊆ F(H)⊆ F(C). Hence C ⊆ F(C).Then F(C)⊆ F
(F(C)
). By definition of C, F(C)⊆ C. Thus
C = F(C).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 60: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/60.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Cantor-Schroder-Bernstein Theorem. Let A and Bbe sets so that there is an injective function f : A → B and aninjective function g : B → A. Then there is a bijective functionh : A → B.
Proof. Define F(X) := A\g[B\ f [X]
]for all X ⊆ A. Then
X ⊆ Y implies f [X]⊆ f [Y], B\ f [X]⊇ B\ f [Y],g[B\ f [X]
]⊇ g
[B\ f [Y]
],
F(X) = A\g[B\ f [X]
]⊆ A\g
[B\ f [Y]
]
= F(Y).Let C :=
⋃{H ∈P(A) : H ⊆ F(H)
}and let c ∈ C. Then there
is an H ∈P(A) with c ∈ H ⊆ F(H)⊆ F(C). Hence C ⊆ F(C).Then F(C)⊆ F
(F(C)
). By definition of C, F(C)⊆ C. Thus
C = F(C).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 61: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/61.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Cantor-Schroder-Bernstein Theorem. Let A and Bbe sets so that there is an injective function f : A → B and aninjective function g : B → A. Then there is a bijective functionh : A → B.
Proof. Define F(X) := A\g[B\ f [X]
]for all X ⊆ A. Then
X ⊆ Y implies f [X]⊆ f [Y], B\ f [X]⊇ B\ f [Y],g[B\ f [X]
]⊇ g
[B\ f [Y]
],
F(X) = A\g[B\ f [X]
]⊆ A\g
[B\ f [Y]
]= F(Y).
Let C :=⋃{
H ∈P(A) : H ⊆ F(H)}
and let c ∈ C. Then thereis an H ∈P(A) with c ∈ H ⊆ F(H)⊆ F(C). Hence C ⊆ F(C).Then F(C)⊆ F
(F(C)
). By definition of C, F(C)⊆ C. Thus
C = F(C).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 62: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/62.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Cantor-Schroder-Bernstein Theorem. Let A and Bbe sets so that there is an injective function f : A → B and aninjective function g : B → A. Then there is a bijective functionh : A → B.
Proof. Define F(X) := A\g[B\ f [X]
]for all X ⊆ A. Then
X ⊆ Y implies f [X]⊆ f [Y], B\ f [X]⊇ B\ f [Y],g[B\ f [X]
]⊇ g
[B\ f [Y]
],
F(X) = A\g[B\ f [X]
]⊆ A\g
[B\ f [Y]
]= F(Y).
Let C :=⋃{
H ∈P(A) : H ⊆ F(H)}
and let c ∈ C. Then thereis an H ∈P(A) with c ∈ H ⊆ F(H)⊆ F(C). Hence C ⊆ F(C).Then F(C)⊆ F
(F(C)
). By definition of C, F(C)⊆ C. Thus
C = F(C).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 63: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/63.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Cantor-Schroder-Bernstein Theorem. Let A and Bbe sets so that there is an injective function f : A → B and aninjective function g : B → A. Then there is a bijective functionh : A → B.
Proof. Define F(X) := A\g[B\ f [X]
]for all X ⊆ A. Then
X ⊆ Y implies f [X]⊆ f [Y], B\ f [X]⊇ B\ f [Y],g[B\ f [X]
]⊇ g
[B\ f [Y]
],
F(X) = A\g[B\ f [X]
]⊆ A\g
[B\ f [Y]
]= F(Y).
Let C :=⋃{
H ∈P(A) : H ⊆ F(H)}
and let c ∈ C.
Then thereis an H ∈P(A) with c ∈ H ⊆ F(H)⊆ F(C). Hence C ⊆ F(C).Then F(C)⊆ F
(F(C)
). By definition of C, F(C)⊆ C. Thus
C = F(C).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 64: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/64.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Cantor-Schroder-Bernstein Theorem. Let A and Bbe sets so that there is an injective function f : A → B and aninjective function g : B → A. Then there is a bijective functionh : A → B.
Proof. Define F(X) := A\g[B\ f [X]
]for all X ⊆ A. Then
X ⊆ Y implies f [X]⊆ f [Y], B\ f [X]⊇ B\ f [Y],g[B\ f [X]
]⊇ g
[B\ f [Y]
],
F(X) = A\g[B\ f [X]
]⊆ A\g
[B\ f [Y]
]= F(Y).
Let C :=⋃{
H ∈P(A) : H ⊆ F(H)}
and let c ∈ C. Then thereis an H ∈P(A) with c ∈ H ⊆ F(H)⊆ F(C).
Hence C ⊆ F(C).Then F(C)⊆ F
(F(C)
). By definition of C, F(C)⊆ C. Thus
C = F(C).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 65: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/65.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Cantor-Schroder-Bernstein Theorem. Let A and Bbe sets so that there is an injective function f : A → B and aninjective function g : B → A. Then there is a bijective functionh : A → B.
Proof. Define F(X) := A\g[B\ f [X]
]for all X ⊆ A. Then
X ⊆ Y implies f [X]⊆ f [Y], B\ f [X]⊇ B\ f [Y],g[B\ f [X]
]⊇ g
[B\ f [Y]
],
F(X) = A\g[B\ f [X]
]⊆ A\g
[B\ f [Y]
]= F(Y).
Let C :=⋃{
H ∈P(A) : H ⊆ F(H)}
and let c ∈ C. Then thereis an H ∈P(A) with c ∈ H ⊆ F(H)⊆ F(C). Hence C ⊆ F(C).
Then F(C)⊆ F(F(C)
). By definition of C, F(C)⊆ C. Thus
C = F(C).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 66: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/66.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Cantor-Schroder-Bernstein Theorem. Let A and Bbe sets so that there is an injective function f : A → B and aninjective function g : B → A. Then there is a bijective functionh : A → B.
Proof. Define F(X) := A\g[B\ f [X]
]for all X ⊆ A. Then
X ⊆ Y implies f [X]⊆ f [Y], B\ f [X]⊇ B\ f [Y],g[B\ f [X]
]⊇ g
[B\ f [Y]
],
F(X) = A\g[B\ f [X]
]⊆ A\g
[B\ f [Y]
]= F(Y).
Let C :=⋃{
H ∈P(A) : H ⊆ F(H)}
and let c ∈ C. Then thereis an H ∈P(A) with c ∈ H ⊆ F(H)⊆ F(C). Hence C ⊆ F(C).Then F(C)⊆ F
(F(C)
).
By definition of C, F(C)⊆ C. ThusC = F(C).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Cantor-Schroder-Bernstein Theorem. Let A and Bbe sets so that there is an injective function f : A → B and aninjective function g : B → A. Then there is a bijective functionh : A → B.
Proof. Define F(X) := A\g[B\ f [X]
]for all X ⊆ A. Then
X ⊆ Y implies f [X]⊆ f [Y], B\ f [X]⊇ B\ f [Y],g[B\ f [X]
]⊇ g
[B\ f [Y]
],
F(X) = A\g[B\ f [X]
]⊆ A\g
[B\ f [Y]
]= F(Y).
Let C :=⋃{
H ∈P(A) : H ⊆ F(H)}
and let c ∈ C. Then thereis an H ∈P(A) with c ∈ H ⊆ F(H)⊆ F(C). Hence C ⊆ F(C).Then F(C)⊆ F
(F(C)
). By definition of C, F(C)⊆ C.
ThusC = F(C).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Cantor-Schroder-Bernstein Theorem. Let A and Bbe sets so that there is an injective function f : A → B and aninjective function g : B → A. Then there is a bijective functionh : A → B.
Proof. Define F(X) := A\g[B\ f [X]
]for all X ⊆ A. Then
X ⊆ Y implies f [X]⊆ f [Y], B\ f [X]⊇ B\ f [Y],g[B\ f [X]
]⊇ g
[B\ f [Y]
],
F(X) = A\g[B\ f [X]
]⊆ A\g
[B\ f [Y]
]= F(Y).
Let C :=⋃{
H ∈P(A) : H ⊆ F(H)}
and let c ∈ C. Then thereis an H ∈P(A) with c ∈ H ⊆ F(H)⊆ F(C). Hence C ⊆ F(C).Then F(C)⊆ F
(F(C)
). By definition of C, F(C)⊆ C. Thus
C = F(C).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.).
C = F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x) 6= f (y) = h(y). If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 70: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/70.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C
= F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x) 6= f (y) = h(y). If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 71: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/71.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C)
= A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x) 6= f (y) = h(y). If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 72: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/72.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C) = A\g[B\ f [C]
]
impliesg[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x) 6= f (y) = h(y). If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 73: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/73.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C
and then B\ f [C] = g−1[A\C]. Henceg−1
∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x) 6= f (y) = h(y). If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 74: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/74.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C].
Henceg−1
∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x) 6= f (y) = h(y). If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 75: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/75.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C].
Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x) 6= f (y) = h(y). If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 76: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/76.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C.
Then h|C : C → f [C] andh|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x) 6= f (y) = h(y). If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 77: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/77.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective.
So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x) 6= f (y) = h(y). If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 78: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/78.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective.
Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x) 6= f (y) = h(y). If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 79: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/79.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y.
If x,y ∈ C,then h(x) = f (x) 6= f (y) = h(y). If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 80: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/80.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x)
= f (x) 6= f (y) = h(y). If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 81: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/81.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x)
6= f (y) = h(y). If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 82: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/82.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x) 6=
f (y) = h(y). If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 83: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/83.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x) 6= f (y)
= h(y). If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 84: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/84.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x) 6= f (y) = h(y).
If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x) 6= f (y) = h(y). If x,y 6∈ C, thenh(x)
= g−1(x) 6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 86: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/86.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x) 6= f (y) = h(y). If x,y 6∈ C, thenh(x) = g−1(x)
6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 87: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/87.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x) 6= f (y) = h(y). If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y)
= h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 88: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/88.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x) 6= f (y) = h(y). If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y) = h(y).
Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 89: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/89.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x) 6= f (y) = h(y). If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C.
Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 90: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/90.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x) 6= f (y) = h(y). If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C].
So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 91: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/91.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x) 6= f (y) = h(y). If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 92: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/92.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Proof (concl.). C = F(C) = A\g[B\ f [C]
]implies
g[B\ f [C]
]= A\C and then B\ f [C] = g−1[A\C]. Hence
g−1∣∣A\C is bijective from A\C onto B\ f [C]. Define h : A → B
by h|C := f |C and h|A\C := g−1∣∣A\C. Then h|C : C → f [C] and
h|A\C : A\C → B\ f [C] are bijective. So h is surjective. Toprove that h is injective, let x,y ∈ A be so that x 6= y. If x,y ∈ C,then h(x) = f (x) 6= f (y) = h(y). If x,y 6∈ C, thenh(x) = g−1(x) 6= g−1(y) = h(y). Otherwise, WLOG x ∈ C andy 6∈ C. Then h(x) ∈ f [C] and h(y) ∈ B\ f [C]. So h(x) 6= h(y).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 93: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/93.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem.
Let A be an infinite set. Then A×A is equivalent toA.
Sketch of proof. Apply Zorn’s Lemma to the set F of allbijective functions f : X×X → X, where X ⊆ A. To prove that amaximal element f : Y×Y → Y of F must be a bijectivefunction from A×A to A, assume that Y is not equivalent to A.There must be an injective function from Y to A\Y . LetZ ⊆ A\Y be equivalent to Y and expand f to a function from(Y ∪Z)× (Y ∪Z) to Y ∪Z, using that Y×Z∪Z×Z∪Z×Y isequivalent to Z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 94: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/94.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let A be an infinite set.
Then A×A is equivalent toA.
Sketch of proof. Apply Zorn’s Lemma to the set F of allbijective functions f : X×X → X, where X ⊆ A. To prove that amaximal element f : Y×Y → Y of F must be a bijectivefunction from A×A to A, assume that Y is not equivalent to A.There must be an injective function from Y to A\Y . LetZ ⊆ A\Y be equivalent to Y and expand f to a function from(Y ∪Z)× (Y ∪Z) to Y ∪Z, using that Y×Z∪Z×Z∪Z×Y isequivalent to Z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 95: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/95.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let A be an infinite set. Then A×A is equivalent toA.
Sketch of proof. Apply Zorn’s Lemma to the set F of allbijective functions f : X×X → X, where X ⊆ A. To prove that amaximal element f : Y×Y → Y of F must be a bijectivefunction from A×A to A, assume that Y is not equivalent to A.There must be an injective function from Y to A\Y . LetZ ⊆ A\Y be equivalent to Y and expand f to a function from(Y ∪Z)× (Y ∪Z) to Y ∪Z, using that Y×Z∪Z×Z∪Z×Y isequivalent to Z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 96: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/96.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let A be an infinite set. Then A×A is equivalent toA.
Sketch of proof.
Apply Zorn’s Lemma to the set F of allbijective functions f : X×X → X, where X ⊆ A. To prove that amaximal element f : Y×Y → Y of F must be a bijectivefunction from A×A to A, assume that Y is not equivalent to A.There must be an injective function from Y to A\Y . LetZ ⊆ A\Y be equivalent to Y and expand f to a function from(Y ∪Z)× (Y ∪Z) to Y ∪Z, using that Y×Z∪Z×Z∪Z×Y isequivalent to Z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 97: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/97.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let A be an infinite set. Then A×A is equivalent toA.
Sketch of proof. Apply Zorn’s Lemma to the set F of allbijective functions f : X×X → X, where X ⊆ A.
To prove that amaximal element f : Y×Y → Y of F must be a bijectivefunction from A×A to A, assume that Y is not equivalent to A.There must be an injective function from Y to A\Y . LetZ ⊆ A\Y be equivalent to Y and expand f to a function from(Y ∪Z)× (Y ∪Z) to Y ∪Z, using that Y×Z∪Z×Z∪Z×Y isequivalent to Z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 98: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/98.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let A be an infinite set. Then A×A is equivalent toA.
Sketch of proof. Apply Zorn’s Lemma to the set F of allbijective functions f : X×X → X, where X ⊆ A. To prove that amaximal element f : Y×Y → Y of F must be a bijectivefunction from A×A to A
, assume that Y is not equivalent to A.There must be an injective function from Y to A\Y . LetZ ⊆ A\Y be equivalent to Y and expand f to a function from(Y ∪Z)× (Y ∪Z) to Y ∪Z, using that Y×Z∪Z×Z∪Z×Y isequivalent to Z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 99: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/99.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let A be an infinite set. Then A×A is equivalent toA.
Sketch of proof. Apply Zorn’s Lemma to the set F of allbijective functions f : X×X → X, where X ⊆ A. To prove that amaximal element f : Y×Y → Y of F must be a bijectivefunction from A×A to A, assume that Y is not equivalent to A.
There must be an injective function from Y to A\Y . LetZ ⊆ A\Y be equivalent to Y and expand f to a function from(Y ∪Z)× (Y ∪Z) to Y ∪Z, using that Y×Z∪Z×Z∪Z×Y isequivalent to Z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 100: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/100.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let A be an infinite set. Then A×A is equivalent toA.
Sketch of proof. Apply Zorn’s Lemma to the set F of allbijective functions f : X×X → X, where X ⊆ A. To prove that amaximal element f : Y×Y → Y of F must be a bijectivefunction from A×A to A, assume that Y is not equivalent to A.There must be an injective function from Y to A\Y .
LetZ ⊆ A\Y be equivalent to Y and expand f to a function from(Y ∪Z)× (Y ∪Z) to Y ∪Z, using that Y×Z∪Z×Z∪Z×Y isequivalent to Z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 101: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/101.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let A be an infinite set. Then A×A is equivalent toA.
Sketch of proof. Apply Zorn’s Lemma to the set F of allbijective functions f : X×X → X, where X ⊆ A. To prove that amaximal element f : Y×Y → Y of F must be a bijectivefunction from A×A to A, assume that Y is not equivalent to A.There must be an injective function from Y to A\Y . LetZ ⊆ A\Y be equivalent to Y and expand f to a function from(Y ∪Z)× (Y ∪Z) to Y ∪Z
, using that Y×Z∪Z×Z∪Z×Y isequivalent to Z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 102: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/102.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let A be an infinite set. Then A×A is equivalent toA.
Sketch of proof. Apply Zorn’s Lemma to the set F of allbijective functions f : X×X → X, where X ⊆ A. To prove that amaximal element f : Y×Y → Y of F must be a bijectivefunction from A×A to A, assume that Y is not equivalent to A.There must be an injective function from Y to A\Y . LetZ ⊆ A\Y be equivalent to Y and expand f to a function from(Y ∪Z)× (Y ∪Z) to Y ∪Z, using that Y×Z∪Z×Z∪Z×Y isequivalent to Z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 103: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/103.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let A be an infinite set. Then A×A is equivalent toA.
Sketch of proof. Apply Zorn’s Lemma to the set F of allbijective functions f : X×X → X, where X ⊆ A. To prove that amaximal element f : Y×Y → Y of F must be a bijectivefunction from A×A to A, assume that Y is not equivalent to A.There must be an injective function from Y to A\Y . LetZ ⊆ A\Y be equivalent to Y and expand f to a function from(Y ∪Z)× (Y ∪Z) to Y ∪Z, using that Y×Z∪Z×Z∪Z×Y isequivalent to Z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 104: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/104.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Definition.
Cardinal arithmetic. Let α and β be cardinalnumbers and let A and B be sets with |A|= α and |B|= β .
1. α +β :=∣∣A×{0}∪B×{1}
∣∣2. αβ := |A×B|3. α
β :=∣∣AB∣∣, where AB := {f : f is a function from A to B}.
Theorem. Let α,β be cardinal numbers so that one of α and β
is infinite. Then α +β = max{α,β} and αβ = max{α,β}.
Proof. Good exercise.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 105: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/105.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Definition. Cardinal arithmetic.
Let α and β be cardinalnumbers and let A and B be sets with |A|= α and |B|= β .
1. α +β :=∣∣A×{0}∪B×{1}
∣∣2. αβ := |A×B|3. α
β :=∣∣AB∣∣, where AB := {f : f is a function from A to B}.
Theorem. Let α,β be cardinal numbers so that one of α and β
is infinite. Then α +β = max{α,β} and αβ = max{α,β}.
Proof. Good exercise.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 106: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/106.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Definition. Cardinal arithmetic. Let α and β be cardinalnumbers and let A and B be sets with |A|= α and |B|= β .
1. α +β :=∣∣A×{0}∪B×{1}
∣∣2. αβ := |A×B|3. α
β :=∣∣AB∣∣, where AB := {f : f is a function from A to B}.
Theorem. Let α,β be cardinal numbers so that one of α and β
is infinite. Then α +β = max{α,β} and αβ = max{α,β}.
Proof. Good exercise.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 107: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/107.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Definition. Cardinal arithmetic. Let α and β be cardinalnumbers and let A and B be sets with |A|= α and |B|= β .
1. α +β :=∣∣A×{0}∪B×{1}
∣∣
2. αβ := |A×B|3. α
β :=∣∣AB∣∣, where AB := {f : f is a function from A to B}.
Theorem. Let α,β be cardinal numbers so that one of α and β
is infinite. Then α +β = max{α,β} and αβ = max{α,β}.
Proof. Good exercise.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 108: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/108.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
Definition. Cardinal arithmetic. Let α and β be cardinalnumbers and let A and B be sets with |A|= α and |B|= β .
1. α +β :=∣∣A×{0}∪B×{1}
∣∣2. αβ := |A×B|
3. αβ :=
∣∣AB∣∣, where AB := {f : f is a function from A to B}.
Theorem. Let α,β be cardinal numbers so that one of α and β
is infinite. Then α +β = max{α,β} and αβ = max{α,β}.
Proof. Good exercise.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 109: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/109.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Definition. Cardinal arithmetic. Let α and β be cardinalnumbers and let A and B be sets with |A|= α and |B|= β .
1. α +β :=∣∣A×{0}∪B×{1}
∣∣2. αβ := |A×B|3. α
β :=∣∣AB∣∣, where AB := {f : f is a function from A to B}.
Theorem. Let α,β be cardinal numbers so that one of α and β
is infinite. Then α +β = max{α,β} and αβ = max{α,β}.
Proof. Good exercise.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 110: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/110.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Definition. Cardinal arithmetic. Let α and β be cardinalnumbers and let A and B be sets with |A|= α and |B|= β .
1. α +β :=∣∣A×{0}∪B×{1}
∣∣2. αβ := |A×B|3. α
β :=∣∣AB∣∣, where AB := {f : f is a function from A to B}.
Theorem.
Let α,β be cardinal numbers so that one of α and β
is infinite. Then α +β = max{α,β} and αβ = max{α,β}.
Proof. Good exercise.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 111: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/111.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Definition. Cardinal arithmetic. Let α and β be cardinalnumbers and let A and B be sets with |A|= α and |B|= β .
1. α +β :=∣∣A×{0}∪B×{1}
∣∣2. αβ := |A×B|3. α
β :=∣∣AB∣∣, where AB := {f : f is a function from A to B}.
Theorem. Let α,β be cardinal numbers so that one of α and β
is infinite.
Then α +β = max{α,β} and αβ = max{α,β}.
Proof. Good exercise.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 112: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/112.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Definition. Cardinal arithmetic. Let α and β be cardinalnumbers and let A and B be sets with |A|= α and |B|= β .
1. α +β :=∣∣A×{0}∪B×{1}
∣∣2. αβ := |A×B|3. α
β :=∣∣AB∣∣, where AB := {f : f is a function from A to B}.
Theorem. Let α,β be cardinal numbers so that one of α and β
is infinite. Then α +β = max{α,β}
and αβ = max{α,β}.
Proof. Good exercise.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 113: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/113.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Definition. Cardinal arithmetic. Let α and β be cardinalnumbers and let A and B be sets with |A|= α and |B|= β .
1. α +β :=∣∣A×{0}∪B×{1}
∣∣2. αβ := |A×B|3. α
β :=∣∣AB∣∣, where AB := {f : f is a function from A to B}.
Theorem. Let α,β be cardinal numbers so that one of α and β
is infinite. Then α +β = max{α,β} and αβ = max{α,β}.
Proof. Good exercise.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 114: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/114.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Definition. Cardinal arithmetic. Let α and β be cardinalnumbers and let A and B be sets with |A|= α and |B|= β .
1. α +β :=∣∣A×{0}∪B×{1}
∣∣2. αβ := |A×B|3. α
β :=∣∣AB∣∣, where AB := {f : f is a function from A to B}.
Theorem. Let α,β be cardinal numbers so that one of α and β
is infinite. Then α +β = max{α,β} and αβ = max{α,β}.
Proof.
Good exercise.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 115: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/115.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Definition. Cardinal arithmetic. Let α and β be cardinalnumbers and let A and B be sets with |A|= α and |B|= β .
1. α +β :=∣∣A×{0}∪B×{1}
∣∣2. αβ := |A×B|3. α
β :=∣∣AB∣∣, where AB := {f : f is a function from A to B}.
Theorem. Let α,β be cardinal numbers so that one of α and β
is infinite. Then α +β = max{α,β} and αβ = max{α,β}.
Proof. Good exercise.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 116: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/116.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Definition. Cardinal arithmetic. Let α and β be cardinalnumbers and let A and B be sets with |A|= α and |B|= β .
1. α +β :=∣∣A×{0}∪B×{1}
∣∣2. αβ := |A×B|3. α
β :=∣∣AB∣∣, where AB := {f : f is a function from A to B}.
Theorem. Let α,β be cardinal numbers so that one of α and β
is infinite. Then α +β = max{α,β} and αβ = max{α,β}.
Proof. Good exercise.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 117: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/117.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem.
Let α,β and γ be cardinal numbers. Then1. αβ αγ = αβ+γ
2. αγβ γ = (αβ )γ
3.(
αβ
)γ
= αβγ
Proof (part 1 only). Let A,B,C be disjoint sets with |A|= α ,|B|= β and |C|= γ . Then AB∪C is equivalent to the set AB×AC
via f ∈ AB∪C 7→ (f |B, f |C). Henceα
β+γ =∣∣AB∪C∣∣ =
∣∣AB×AC∣∣ = αβ
αγ .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 118: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/118.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let α,β and γ be cardinal numbers.
Then1. αβ αγ = αβ+γ
2. αγβ γ = (αβ )γ
3.(
αβ
)γ
= αβγ
Proof (part 1 only). Let A,B,C be disjoint sets with |A|= α ,|B|= β and |C|= γ . Then AB∪C is equivalent to the set AB×AC
via f ∈ AB∪C 7→ (f |B, f |C). Henceα
β+γ =∣∣AB∪C∣∣ =
∣∣AB×AC∣∣ = αβ
αγ .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 119: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/119.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let α,β and γ be cardinal numbers. Then1. αβ αγ = αβ+γ
2. αγβ γ = (αβ )γ
3.(
αβ
)γ
= αβγ
Proof (part 1 only). Let A,B,C be disjoint sets with |A|= α ,|B|= β and |C|= γ . Then AB∪C is equivalent to the set AB×AC
via f ∈ AB∪C 7→ (f |B, f |C). Henceα
β+γ =∣∣AB∪C∣∣ =
∣∣AB×AC∣∣ = αβ
αγ .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 120: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/120.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let α,β and γ be cardinal numbers. Then1. αβ αγ = αβ+γ
2. αγβ γ = (αβ )γ
3.(
αβ
)γ
= αβγ
Proof (part 1 only). Let A,B,C be disjoint sets with |A|= α ,|B|= β and |C|= γ . Then AB∪C is equivalent to the set AB×AC
via f ∈ AB∪C 7→ (f |B, f |C). Henceα
β+γ =∣∣AB∪C∣∣ =
∣∣AB×AC∣∣ = αβ
αγ .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 121: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/121.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let α,β and γ be cardinal numbers. Then1. αβ αγ = αβ+γ
2. αγβ γ = (αβ )γ
3.(
αβ
)γ
= αβγ
Proof (part 1 only). Let A,B,C be disjoint sets with |A|= α ,|B|= β and |C|= γ . Then AB∪C is equivalent to the set AB×AC
via f ∈ AB∪C 7→ (f |B, f |C). Henceα
β+γ =∣∣AB∪C∣∣ =
∣∣AB×AC∣∣ = αβ
αγ .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 122: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/122.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let α,β and γ be cardinal numbers. Then1. αβ αγ = αβ+γ
2. αγβ γ = (αβ )γ
3.(
αβ
)γ
= αβγ
Proof (part 1 only).
Let A,B,C be disjoint sets with |A|= α ,|B|= β and |C|= γ . Then AB∪C is equivalent to the set AB×AC
via f ∈ AB∪C 7→ (f |B, f |C). Henceα
β+γ =∣∣AB∪C∣∣ =
∣∣AB×AC∣∣ = αβ
αγ .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 123: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/123.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let α,β and γ be cardinal numbers. Then1. αβ αγ = αβ+γ
2. αγβ γ = (αβ )γ
3.(
αβ
)γ
= αβγ
Proof (part 1 only). Let A,B,C be disjoint sets with |A|= α ,|B|= β and |C|= γ .
Then AB∪C is equivalent to the set AB×AC
via f ∈ AB∪C 7→ (f |B, f |C). Henceα
β+γ =∣∣AB∪C∣∣ =
∣∣AB×AC∣∣ = αβ
αγ .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 124: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/124.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let α,β and γ be cardinal numbers. Then1. αβ αγ = αβ+γ
2. αγβ γ = (αβ )γ
3.(
αβ
)γ
= αβγ
Proof (part 1 only). Let A,B,C be disjoint sets with |A|= α ,|B|= β and |C|= γ . Then AB∪C is equivalent to the set AB×AC
via f ∈ AB∪C 7→ (f |B, f |C). Henceα
β+γ =∣∣AB∪C∣∣ =
∣∣AB×AC∣∣ = αβ
αγ .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 125: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/125.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let α,β and γ be cardinal numbers. Then1. αβ αγ = αβ+γ
2. αγβ γ = (αβ )γ
3.(
αβ
)γ
= αβγ
Proof (part 1 only). Let A,B,C be disjoint sets with |A|= α ,|B|= β and |C|= γ . Then AB∪C is equivalent to the set AB×AC
via f ∈ AB∪C
7→ (f |B, f |C). Henceα
β+γ =∣∣AB∪C∣∣ =
∣∣AB×AC∣∣ = αβ
αγ .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 126: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/126.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let α,β and γ be cardinal numbers. Then1. αβ αγ = αβ+γ
2. αγβ γ = (αβ )γ
3.(
αβ
)γ
= αβγ
Proof (part 1 only). Let A,B,C be disjoint sets with |A|= α ,|B|= β and |C|= γ . Then AB∪C is equivalent to the set AB×AC
via f ∈ AB∪C 7→ (f |B, f |C).
Henceα
β+γ =∣∣AB∪C∣∣ =
∣∣AB×AC∣∣ = αβ
αγ .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 127: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/127.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let α,β and γ be cardinal numbers. Then1. αβ αγ = αβ+γ
2. αγβ γ = (αβ )γ
3.(
αβ
)γ
= αβγ
Proof (part 1 only). Let A,B,C be disjoint sets with |A|= α ,|B|= β and |C|= γ . Then AB∪C is equivalent to the set AB×AC
via f ∈ AB∪C 7→ (f |B, f |C). Henceα
β+γ
=∣∣AB∪C∣∣ =
∣∣AB×AC∣∣ = αβ
αγ .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 128: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/128.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let α,β and γ be cardinal numbers. Then1. αβ αγ = αβ+γ
2. αγβ γ = (αβ )γ
3.(
αβ
)γ
= αβγ
Proof (part 1 only). Let A,B,C be disjoint sets with |A|= α ,|B|= β and |C|= γ . Then AB∪C is equivalent to the set AB×AC
via f ∈ AB∪C 7→ (f |B, f |C). Henceα
β+γ =∣∣AB∪C∣∣
=∣∣AB×AC∣∣ = α
βα
γ .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 129: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/129.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let α,β and γ be cardinal numbers. Then1. αβ αγ = αβ+γ
2. αγβ γ = (αβ )γ
3.(
αβ
)γ
= αβγ
Proof (part 1 only). Let A,B,C be disjoint sets with |A|= α ,|B|= β and |C|= γ . Then AB∪C is equivalent to the set AB×AC
via f ∈ AB∪C 7→ (f |B, f |C). Henceα
β+γ =∣∣AB∪C∣∣ =
∣∣AB×AC∣∣
= αβ
αγ .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 130: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/130.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let α,β and γ be cardinal numbers. Then1. αβ αγ = αβ+γ
2. αγβ γ = (αβ )γ
3.(
αβ
)γ
= αβγ
Proof (part 1 only). Let A,B,C be disjoint sets with |A|= α ,|B|= β and |C|= γ . Then AB∪C is equivalent to the set AB×AC
via f ∈ AB∪C 7→ (f |B, f |C). Henceα
β+γ =∣∣AB∪C∣∣ =
∣∣AB×AC∣∣ = αβ
αγ .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 131: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/131.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
Theorem. Let α,β and γ be cardinal numbers. Then1. αβ αγ = αβ+γ
2. αγβ γ = (αβ )γ
3.(
αβ
)γ
= αβγ
Proof (part 1 only). Let A,B,C be disjoint sets with |A|= α ,|B|= β and |C|= γ . Then AB∪C is equivalent to the set AB×AC
via f ∈ AB∪C 7→ (f |B, f |C). Henceα
β+γ =∣∣AB∪C∣∣ =
∣∣AB×AC∣∣ = αβ
αγ .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 132: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/132.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
A Mysterious Gap
ℵ0 denotes the first infinite cardinal number. That is,ℵ0 = ω = N0.We know that 2ℵ0 is not countable.But is it equal to the first uncountable ordinal ℵ1?
Axiom. The Continuum Hypothesis. 2ℵ0 = ℵ1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 133: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/133.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
A Mysterious Gapℵ0 denotes the first infinite cardinal number.
That is,ℵ0 = ω = N0.We know that 2ℵ0 is not countable.But is it equal to the first uncountable ordinal ℵ1?
Axiom. The Continuum Hypothesis. 2ℵ0 = ℵ1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 134: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/134.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
A Mysterious Gapℵ0 denotes the first infinite cardinal number. That is,ℵ0 = ω = N0.
We know that 2ℵ0 is not countable.But is it equal to the first uncountable ordinal ℵ1?
Axiom. The Continuum Hypothesis. 2ℵ0 = ℵ1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 135: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/135.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
A Mysterious Gapℵ0 denotes the first infinite cardinal number. That is,ℵ0 = ω = N0.We know that 2ℵ0 is not countable.
But is it equal to the first uncountable ordinal ℵ1?
Axiom. The Continuum Hypothesis. 2ℵ0 = ℵ1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 136: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/136.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
A Mysterious Gapℵ0 denotes the first infinite cardinal number. That is,ℵ0 = ω = N0.We know that 2ℵ0 is not countable.But is it equal to the first uncountable ordinal ℵ1?
Axiom. The Continuum Hypothesis. 2ℵ0 = ℵ1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 137: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/137.jpg)
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Finite Sizes Infinite Sizes Cardinal Arithmetic
A Mysterious Gapℵ0 denotes the first infinite cardinal number. That is,ℵ0 = ω = N0.We know that 2ℵ0 is not countable.But is it equal to the first uncountable ordinal ℵ1?
Axiom.
The Continuum Hypothesis. 2ℵ0 = ℵ1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 138: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/138.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
A Mysterious Gapℵ0 denotes the first infinite cardinal number. That is,ℵ0 = ω = N0.We know that 2ℵ0 is not countable.But is it equal to the first uncountable ordinal ℵ1?
Axiom. The Continuum Hypothesis.
2ℵ0 = ℵ1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 139: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/139.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
A Mysterious Gapℵ0 denotes the first infinite cardinal number. That is,ℵ0 = ω = N0.We know that 2ℵ0 is not countable.But is it equal to the first uncountable ordinal ℵ1?
Axiom. The Continuum Hypothesis. 2ℵ0 = ℵ1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis
![Page 140: Cardinal Numbers and the Continuum Hypothesis · Finite Sizes Infinite Sizes Cardinal Arithmetic Introduction 1. We want a standard “size” for each set, just like the number](https://reader033.vdocuments.us/reader033/viewer/2022050414/5f8a6994e57aa113a60e6418/html5/thumbnails/140.jpg)
logo1
Finite Sizes Infinite Sizes Cardinal Arithmetic
A Mysterious Gapℵ0 denotes the first infinite cardinal number. That is,ℵ0 = ω = N0.We know that 2ℵ0 is not countable.But is it equal to the first uncountable ordinal ℵ1?
Axiom. The Continuum Hypothesis. 2ℵ0 = ℵ1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cardinal Numbers and the Continuum Hypothesis