Discrete MathematicsMathematical Induction

Math 245January 29, 2013

Undefined terms and axioms

Every logical system must contain a (hopefully small) collection ofterms that remain undefined.

For example, the terms “set” and “element” are usually left undefined.

Similarly, every logical system must contain at least a few statements,called axioms or postulates, which we accept as true without proof.

For example, the defining axiom of Euclidean geometry is the ParallelPostulate.

To understand a proof by induction, we have to understand theaxioms of the natural numbers.

These are the Peano Axioms

Undefined terms and axioms

Every logical system must contain a (hopefully small) collection ofterms that remain undefined.

For example, the terms “set” and “element” are usually left undefined.

Similarly, every logical system must contain at least a few statements,called axioms or postulates, which we accept as true without proof.

For example, the defining axiom of Euclidean geometry is the ParallelPostulate.

To understand a proof by induction, we have to understand theaxioms of the natural numbers.

These are the Peano Axioms

Undefined terms and axioms

Every logical system must contain a (hopefully small) collection ofterms that remain undefined.

For example, the terms “set” and “element” are usually left undefined.

Similarly, every logical system must contain at least a few statements,called axioms or postulates, which we accept as true without proof.

For example, the defining axiom of Euclidean geometry is the ParallelPostulate.

To understand a proof by induction, we have to understand theaxioms of the natural numbers.

These are the Peano Axioms

Undefined terms and axioms

Every logical system must contain a (hopefully small) collection ofterms that remain undefined.

For example, the terms “set” and “element” are usually left undefined.

Similarly, every logical system must contain at least a few statements,called axioms or postulates, which we accept as true without proof.

For example, the defining axiom of Euclidean geometry is the ParallelPostulate.

To understand a proof by induction, we have to understand theaxioms of the natural numbers.

These are the Peano Axioms

Undefined terms and axioms

Every logical system must contain a (hopefully small) collection ofterms that remain undefined.

For example, the terms “set” and “element” are usually left undefined.

Similarly, every logical system must contain at least a few statements,called axioms or postulates, which we accept as true without proof.

For example, the defining axiom of Euclidean geometry is the ParallelPostulate.

To understand a proof by induction, we have to understand theaxioms of the natural numbers.

These are the Peano Axioms

Undefined terms and axioms

Every logical system must contain a (hopefully small) collection ofterms that remain undefined.

For example, the terms “set” and “element” are usually left undefined.

Similarly, every logical system must contain at least a few statements,called axioms or postulates, which we accept as true without proof.

For example, the defining axiom of Euclidean geometry is the ParallelPostulate.

To understand a proof by induction, we have to understand theaxioms of the natural numbers.

These are the Peano Axioms

Undefined terms and axioms

Every logical system must contain a (hopefully small) collection ofterms that remain undefined.

For example, the terms “set” and “element” are usually left undefined.

Similarly, every logical system must contain at least a few statements,called axioms or postulates, which we accept as true without proof.

For example, the defining axiom of Euclidean geometry is the ParallelPostulate.

To understand a proof by induction, we have to understand theaxioms of the natural numbers.

These are the Peano Axioms

Peano Axioms

The natural number system is a set Z+, an element 1 of Z+, and afunction, s : Z+ → Z+, called the successor operation with thefollowing properties:

P1: 1 6= s(n) for any n in Z+.P2: For any numbers m and n in Z+, if s(m) = s(n), then

m = n.P3: If M is a subset of Z+ that contains 1 and if s(n) ∈ M

for each n ∈ M, then

M = Z+.

These axioms lead to the following theorem, which is the basis forevery proof by “induction”

Peano Axioms

The natural number system is a set Z+, an element 1 of Z+, and afunction, s : Z+ → Z+, called the successor operation with thefollowing properties:

P1: 1 6= s(n) for any n in Z+.P2: For any numbers m and n in Z+, if s(m) = s(n), then

m = n.P3: If M is a subset of Z+ that contains 1 and if s(n) ∈ M

for each n ∈ M, then

M = Z+.

These axioms lead to the following theorem, which is the basis forevery proof by “induction”

Peano Axioms

The natural number system is a set Z+, an element 1 of Z+, and afunction, s : Z+ → Z+, called the successor operation with thefollowing properties:

P1: 1 6= s(n) for any n in Z+.

P2: For any numbers m and n in Z+, if s(m) = s(n), thenm = n.

P3: If M is a subset of Z+ that contains 1 and if s(n) ∈ Mfor each n ∈ M, then

M = Z+.

These axioms lead to the following theorem, which is the basis forevery proof by “induction”

Peano Axioms

The natural number system is a set Z+, an element 1 of Z+, and afunction, s : Z+ → Z+, called the successor operation with thefollowing properties:

P1: 1 6= s(n) for any n in Z+.P2: For any numbers m and n in Z+, if s(m) = s(n), then

m = n.

P3: If M is a subset of Z+ that contains 1 and if s(n) ∈ Mfor each n ∈ M, then

M = Z+.

These axioms lead to the following theorem, which is the basis forevery proof by “induction”

Peano Axioms

The natural number system is a set Z+, an element 1 of Z+, and afunction, s : Z+ → Z+, called the successor operation with thefollowing properties:

P1: 1 6= s(n) for any n in Z+.P2: For any numbers m and n in Z+, if s(m) = s(n), then

m = n.P3: If M is a subset of Z+ that contains 1 and if s(n) ∈ M

for each n ∈ M, then

M = Z+.

These axioms lead to the following theorem, which is the basis forevery proof by “induction”

Peano Axioms

The natural number system is a set Z+, an element 1 of Z+, and afunction, s : Z+ → Z+, called the successor operation with thefollowing properties:

P1: 1 6= s(n) for any n in Z+.P2: For any numbers m and n in Z+, if s(m) = s(n), then

m = n.P3: If M is a subset of Z+ that contains 1 and if s(n) ∈ M

for each n ∈ M, then M = Z+.

These axioms lead to the following theorem, which is the basis forevery proof by “induction”

Peano Axioms

The natural number system is a set Z+, an element 1 of Z+, and afunction, s : Z+ → Z+, called the successor operation with thefollowing properties:

P1: 1 6= s(n) for any n in Z+.P2: For any numbers m and n in Z+, if s(m) = s(n), then

m = n.P3: If M is a subset of Z+ that contains 1 and if s(n) ∈ M

for each n ∈ M, then M = Z+.

These axioms lead to the following theorem, which is the basis forevery proof by “induction”

Theorem

Suppose that M is a subset of the set Z+ of natural numbers with thefollowing two properties:

1 The natural number 1 ∈ M.

2 If the natural number k belongs to M, then the next natural numberk + 1 belongs to M.

Then all natural numbers belong to M, that is M = Z+.

Theorem

Suppose that M is a subset of the set Z+ of natural numbers with thefollowing two properties:

1 The natural number 1 ∈ M.

2 If the natural number k belongs to M, then the next natural numberk + 1 belongs to M.

Then all natural numbers belong to M, that is M = Z+.

Theorem

Suppose that M is a subset of the set Z+ of natural numbers with thefollowing two properties:

1 The natural number 1 ∈ M.

2 If the natural number k belongs to M, then the next natural numberk + 1 belongs to M.

Then all natural numbers belong to M, that is M = Z+.

Theorem

Suppose that M is a subset of the set Z+ of natural numbers with thefollowing two properties:

1 The natural number 1 ∈ M.

2 If the natural number k belongs to M, then the next natural numberk + 1 belongs to M.

Then all natural numbers belong to M, that is M = Z+.

Theorem

Suppose that M is a subset of the set Z+ of natural numbers with thefollowing two properties:

1 The natural number 1 ∈ M.

2 If the natural number k belongs to M, then the next natural numberk + 1 belongs to M.

Then all natural numbers belong to M, that is M = Z+.

Template for a Proof by Induction

Theorem A statement involving (positive) integers.

Proof Our proof is by induction.

Basis: We show that our statement holds for n = 1 (or anothersuitable integer).

Induction hypothesis: We assume that our statement holds for n = k.

(For yourself, make sure you understand what the statement would say forn = k + 1.)

Show that the statement holds for n = k + 1.(You should use the induction hypothesis here.)

By the Principle of Mathematical Induction, we have shown that thestatement holds for all integers greater than or equal to 1.

Template for a Proof by Induction

Theorem A statement involving (positive) integers.

Proof Our proof is by induction.

Basis: We show that our statement holds for n = 1 (or anothersuitable integer).

Induction hypothesis: We assume that our statement holds for n = k.

(For yourself, make sure you understand what the statement would say forn = k + 1.)

Show that the statement holds for n = k + 1.(You should use the induction hypothesis here.)

By the Principle of Mathematical Induction, we have shown that thestatement holds for all integers greater than or equal to 1.

Template for a Proof by Induction

Theorem A statement involving (positive) integers.

Proof Our proof is by induction.

Basis: We show that our statement holds for n = 1 (or anothersuitable integer).

Induction hypothesis: We assume that our statement holds for n = k.

(For yourself, make sure you understand what the statement would say forn = k + 1.)

Show that the statement holds for n = k + 1.(You should use the induction hypothesis here.)

By the Principle of Mathematical Induction, we have shown that thestatement holds for all integers greater than or equal to 1.

Template for a Proof by Induction

Theorem A statement involving (positive) integers.

Proof Our proof is by induction.

Basis: We show that our statement holds for n = 1 (or anothersuitable integer).

Induction hypothesis: We assume that our statement holds for n = k.

(For yourself, make sure you understand what the statement would say forn = k + 1.)

Show that the statement holds for n = k + 1.(You should use the induction hypothesis here.)

By the Principle of Mathematical Induction, we have shown that thestatement holds for all integers greater than or equal to 1.

Template for a Proof by Induction

Theorem A statement involving (positive) integers.

Proof Our proof is by induction.

Basis: We show that our statement holds for n = 1 (or anothersuitable integer).

Induction hypothesis: We assume that our statement holds for n = k.

(For yourself, make sure you understand what the statement would say forn = k + 1.)

Show that the statement holds for n = k + 1.(You should use the induction hypothesis here.)

By the Principle of Mathematical Induction, we have shown that thestatement holds for all integers greater than or equal to 1.

Template for a Proof by Induction

Theorem A statement involving (positive) integers.

Proof Our proof is by induction.

Basis: We show that our statement holds for n = 1 (or anothersuitable integer).

Induction hypothesis: We assume that our statement holds for n = k.

(For yourself, make sure you understand what the statement would say forn = k + 1.)

Show that the statement holds for n = k + 1.(You should use the induction hypothesis here.)

By the Principle of Mathematical Induction, we have shown that thestatement holds for all integers greater than or equal to 1.

Template for a Proof by Induction

Theorem A statement involving (positive) integers.

Proof Our proof is by induction.

Basis: We show that our statement holds for n = 1 (or anothersuitable integer).

Induction hypothesis: We assume that our statement holds for n = k.

(For yourself, make sure you understand what the statement would say forn = k + 1.)

Show that the statement holds for n = k + 1.(You should use the induction hypothesis here.)

By the Principle of Mathematical Induction, we have shown that thestatement holds for all integers greater than or equal to 1.

Template for a Proof by Induction

Theorem A statement involving (positive) integers.

Proof Our proof is by induction.

Basis: We show that our statement holds for n = 1 (or anothersuitable integer).

Induction hypothesis: We assume that our statement holds for n = k.

(For yourself, make sure you understand what the statement would say forn = k + 1.)

Show that the statement holds for n = k + 1.(You should use the induction hypothesis here.)

By the Principle of Mathematical Induction, we have shown that thestatement holds for all integers greater than or equal to 1.

Strong Induction

Theorem Another statement involving (positive) integers.

Proof Our proof is by induction.

Basis: We show that our statement holds for n = 1 (or anothersuitable integer).

Induction hypothesis: We assume that our statement holds for allintegers n = 1, 2, 3, . . . , k .

(For yourself, make sure you understand what the statement would say forn = k + 1.)

Show that the statement holds for n = k + 1.(You should use the induction hypothesis here.)

By the Principle of Mathematical Induction, we have shown that thestatement holds for all integers greater than or equal to 1.

Strong Induction

Theorem Another statement involving (positive) integers.

Proof Our proof is by induction.

Basis: We show that our statement holds for n = 1 (or anothersuitable integer).

Induction hypothesis: We assume that our statement holds for allintegers n = 1, 2, 3, . . . , k .

(For yourself, make sure you understand what the statement would say forn = k + 1.)

Show that the statement holds for n = k + 1.(You should use the induction hypothesis here.)

By the Principle of Mathematical Induction, we have shown that thestatement holds for all integers greater than or equal to 1.

Strong Induction

Theorem Another statement involving (positive) integers.

Proof Our proof is by induction.

Basis: We show that our statement holds for n = 1 (or anothersuitable integer).

Induction hypothesis: We assume that our statement holds for allintegers n = 1, 2, 3, . . . , k .

(For yourself, make sure you understand what the statement would say forn = k + 1.)

Show that the statement holds for n = k + 1.(You should use the induction hypothesis here.)

By the Principle of Mathematical Induction, we have shown that thestatement holds for all integers greater than or equal to 1.

Strong Induction

Theorem Another statement involving (positive) integers.

Proof Our proof is by induction.

Basis: We show that our statement holds for n = 1 (or anothersuitable integer).

Induction hypothesis: We assume that our statement holds for allintegers n = 1, 2, 3, . . . , k .

(For yourself, make sure you understand what the statement would say forn = k + 1.)

Show that the statement holds for n = k + 1.(You should use the induction hypothesis here.)

By the Principle of Mathematical Induction, we have shown that thestatement holds for all integers greater than or equal to 1.

Strong Induction

Theorem Another statement involving (positive) integers.

Proof Our proof is by induction.

Basis: We show that our statement holds for n = 1 (or anothersuitable integer).

Induction hypothesis: We assume that our statement holds for allintegers n = 1, 2, 3, . . . , k .

(For yourself, make sure you understand what the statement would say forn = k + 1.)

Show that the statement holds for n = k + 1.(You should use the induction hypothesis here.)

By the Principle of Mathematical Induction, we have shown that thestatement holds for all integers greater than or equal to 1.

Strong Induction

Theorem Another statement involving (positive) integers.

Proof Our proof is by induction.

Basis: We show that our statement holds for n = 1 (or anothersuitable integer).

Induction hypothesis: We assume that our statement holds for allintegers n = 1, 2, 3, . . . , k .

(For yourself, make sure you understand what the statement would say forn = k + 1.)

Show that the statement holds for n = k + 1.(You should use the induction hypothesis here.)

By the Principle of Mathematical Induction, we have shown that thestatement holds for all integers greater than or equal to 1.

Strong Induction

Theorem Another statement involving (positive) integers.

Proof Our proof is by induction.

Basis: We show that our statement holds for n = 1 (or anothersuitable integer).

Induction hypothesis: We assume that our statement holds for allintegers n = 1, 2, 3, . . . , k .

(For yourself, make sure you understand what the statement would say forn = k + 1.)

Show that the statement holds for n = k + 1.(You should use the induction hypothesis here.)

By the Principle of Mathematical Induction, we have shown that thestatement holds for all integers greater than or equal to 1.

Strong Induction

Theorem Another statement involving (positive) integers.

Proof Our proof is by induction.

Basis: We show that our statement holds for n = 1 (or anothersuitable integer).

Induction hypothesis: We assume that our statement holds for allintegers n = 1, 2, 3, . . . , k .

(For yourself, make sure you understand what the statement would say forn = k + 1.)

Show that the statement holds for n = k + 1.(You should use the induction hypothesis here.)

By the Principle of Mathematical Induction, we have shown that thestatement holds for all integers greater than or equal to 1.