math 1300: section 7-2 sets

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Set Properties and Set NotationSet Operations

Math 1300 Finite MathematicsSection 7-2: Sets

Department of MathematicsUniversity of Missouri

October 16, 2009

Math 1300 Finite Mathematics

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A set is any collection of objects specified in such a waythat we can determine whether a given object is or is not inthe collection.

Capital letters, such as A, B, and C are often used todesignate particular sets.Each object in a set is called a member or element of theset.


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A set is any collection of objects specified in such a waythat we can determine whether a given object is or is not inthe collection.Capital letters, such as A, B, and C are often used todesignate particular sets.

Each object in a set is called a member or element of theset.


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A set is any collection of objects specified in such a waythat we can determine whether a given object is or is not inthe collection.Capital letters, such as A, B, and C are often used todesignate particular sets.Each object in a set is called a member or element of theset.


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Symbolically,

a ∈ A means "a is an element of set A"a 6∈ A means "a is not an element of set A"

A set without any elements is called the empty, or null,set.Symbollically

∅ denotes the empty set


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Symbolically,


A set without any elements is called the empty, or null,set.

Symbollically



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Symbolically,


A set without any elements is called the empty, or null,set.Symbollically



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A set is usually described either by listing all its elementsbetween braces { } or by enclosing a rule within braces thatdetermines the elements of the set.

Thus, if P(x) is a statement about x , then

S = {x |P(x)} means “S is the set of all x such that P(x) is true”

Example:Rule Listing

{x |x is a weekend day} = {Saturday, Sunday}{x |x2 = 4} = {−2, 2}

{x |x is a positive odd number} = {1, 3, 5, . . . }

The first two sets in this example are finite sets; the last set isan infinite set.


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When listing the elements of a set, we do not list anelement more than once, and the order in which theelements are listed does not matter.

A is a subset of B if every element of A is also an elementof B.If A and B have exactly the same elments, then the twosets are said to be equal.Symbollically,A ⊂ B means “A is a subset of B”A = B means “A equals B”A 6⊂ B means “A is not a subset of B”A 6= B means “A and B do not have the same elements”∅ is a subset of every set.


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When listing the elements of a set, we do not list anelement more than once, and the order in which theelements are listed does not matter.A is a subset of B if every element of A is also an elementof B.

If A and B have exactly the same elments, then the twosets are said to be equal.Symbollically,A ⊂ B means “A is a subset of B”A = B means “A equals B”A 6⊂ B means “A is not a subset of B”A 6= B means “A and B do not have the same elements”∅ is a subset of every set.


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When listing the elements of a set, we do not list anelement more than once, and the order in which theelements are listed does not matter.A is a subset of B if every element of A is also an elementof B.If A and B have exactly the same elments, then the twosets are said to be equal.

Symbollically,A ⊂ B means “A is a subset of B”A = B means “A equals B”A 6⊂ B means “A is not a subset of B”A 6= B means “A and B do not have the same elements”∅ is a subset of every set.


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When listing the elements of a set, we do not list anelement more than once, and the order in which theelements are listed does not matter.A is a subset of B if every element of A is also an elementof B.If A and B have exactly the same elments, then the twosets are said to be equal.Symbollically,A ⊂ B means “A is a subset of B”A = B means “A equals B”A 6⊂ B means “A is not a subset of B”A 6= B means “A and B do not have the same elements”

∅ is a subset of every set.


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When listing the elements of a set, we do not list anelement more than once, and the order in which theelements are listed does not matter.A is a subset of B if every element of A is also an elementof B.If A and B have exactly the same elments, then the twosets are said to be equal.Symbollically,A ⊂ B means “A is a subset of B”A = B means “A equals B”A 6⊂ B means “A is not a subset of B”A 6= B means “A and B do not have the same elements”∅ is a subset of every set.


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Example: List all subsets of the set A = {b, c, d}.

First, ∅ ⊂ A.

Also, every set is a subset of itself, so A ⊂ A.

What one-element subsets? {b}, {c} and {d}.

What two-element subsets are there? {b, c}, {b, d}, {c, d}

There are a total of 8 subsets of set A.


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First, ∅ ⊂ A.


What one-element subsets?

{b}, {c} and {d}.




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First, ∅ ⊂ A.






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First, ∅ ⊂ A.



What two-element subsets are there?

{b, c}, {b, d}, {c, d}



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First, ∅ ⊂ A.






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Example: If A = {−3,−1, 1, 3}, B = {3,−3, 1,−1} andC = {−3,−2,−1, 0, 1, 2, 3}, each of the following statements istrue:

A = B A ⊂ C A ⊂ BC 6= A C 6⊂ A B ⊂ A∅ ⊂ A ∅ ⊂ C ∅ 6∈ A


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A = B

A ⊂ C A ⊂ BC 6= A C 6⊂ A B ⊂ A∅ ⊂ A ∅ ⊂ C ∅ 6∈ A


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A = B A ⊂ C

A ⊂ BC 6= A C 6⊂ A B ⊂ A∅ ⊂ A ∅ ⊂ C ∅ 6∈ A


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A = B A ⊂ C A ⊂ B

C 6= A C 6⊂ A B ⊂ A∅ ⊂ A ∅ ⊂ C ∅ 6∈ A


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A = B A ⊂ C A ⊂ BC 6= A

C 6⊂ A B ⊂ A∅ ⊂ A ∅ ⊂ C ∅ 6∈ A


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A = B A ⊂ C A ⊂ BC 6= A C 6⊂ A

B ⊂ A∅ ⊂ A ∅ ⊂ C ∅ 6∈ A


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A = B A ⊂ C A ⊂ BC 6= A C 6⊂ A B ⊂ A

∅ ⊂ A ∅ ⊂ C ∅ 6∈ A


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A = B A ⊂ C A ⊂ BC 6= A C 6⊂ A B ⊂ A∅ ⊂ A

∅ ⊂ C ∅ 6∈ A


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A = B A ⊂ C A ⊂ BC 6= A C 6⊂ A B ⊂ A∅ ⊂ A ∅ ⊂ C

∅ 6∈ A


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A = B A ⊂ C A ⊂ BC 6= A C 6⊂ A B ⊂ A∅ ⊂ A ∅ ⊂ C ∅ 6∈ A


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DefinitionThe union of two sets A and B is the set of all elements formedby combining all the elements of set A and all the elements ofset B into one set. It is written A ∪ B. Symbolically,

A ∪ B = {x |x ∈ A or x ∈ B}

U

A B


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DefinitionThe intersection of two sets A and B is the set of all elementsthat are common to both A and B. It is written A ∩ B.Symbolically,

A ∩ B = {x |x ∈ A and x ∈ B}

U

A B


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If two sets have no elements in common, they are said tobe disjoint. Two sets A and B are disjoint if A ∩ B = ∅.

The set of all elements under consideration is called theuniversal set U. Once the universal set is determined for aparticular problem, all other sets under consideration mustbe subsets of U.


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If two sets have no elements in common, they are said tobe disjoint. Two sets A and B are disjoint if A ∩ B = ∅.The set of all elements under consideration is called theuniversal set U. Once the universal set is determined for aparticular problem, all other sets under consideration mustbe subsets of U.


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DefinitionThe complement of a set A is defined as the set of elementsthat are contained in U, the universal set, but not contained inset A. We donte the complement by A′. Symbolically,

A′ = {x ∈ U|x 6∈ A}

U

A B


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Example: A marketing survey of 1,000 commuters found that600 answered listen to the news, 500 listen to music, and 300listen to both. Let N = set of commuters in the sample wholisten to news and M = set of commuters in the sample wholisten to music. Find the number of commuters in the setN ∩M ′.

The number of elements in a set A is denoted by n(A), so inthis case we are looking for n(N ∩M ′).

The study is based on 1000 commuters, so n(U) = 1000.

The number of elements in the four sections in the Venndiagram need to add up to 1000.


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U = 1, 000

M N

N ∩M ′

300200 300

So, 300 commuters listen to news and not music.


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U = 1, 000

M N

N ∩M ′300

200 300



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U = 1, 000

M N

N ∩M ′300200

300



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U = 1, 000

M N

300200 300



math 1300: section 7-2 sets

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set properties

element of set aa set

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set isan innite set

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