Discrete Mathematics
Mathematical Induction
Math 245
January 29, 2013
Undefined terms and axioms
Undefined terms and axioms
Every logical system must contain a (hopefully small) collection of
terms that remain undefined.
Undefined terms and axioms
Every logical system must contain a (hopefully small) collection of
terms that remain undefined.
For example, the terms “set” and “element” are usually left undefined.
Undefined terms and axioms
Every logical system must contain a (hopefully small) collection of
terms 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.
Undefined terms and axioms
Every logical system must contain a (hopefully small) collection of
terms 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 Parallel
Postulate.
Undefined terms and axioms
Every logical system must contain a (hopefully small) collection of
terms 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 Parallel
Postulate.
To understand a proof by induction, we have to understand the
axioms of the natural numbers.
Undefined terms and axioms
Every logical system must contain a (hopefully small) collection of
terms 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 Parallel
Postulate.
To understand a proof by induction, we have to understand the
axioms of the natural numbers.
These are the Peano Axioms
Peano Axioms
Peano Axioms
The natural number system is a set Z+ , an element 1 of Z+ , and a
function, s : Z+ → Z+ , called the successor operation with the
following properties:
Peano Axioms
The natural number system is a set Z+ , an element 1 of Z+ , and a
function, s : Z+ → Z+ , called the successor operation with the
following properties:
P1: 1 6= s(n) for any n in Z+ .
Peano Axioms
The natural number system is a set Z+ , an element 1 of Z+ , and a
function, s : Z+ → Z+ , called the successor operation with the
following 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.
Peano Axioms
The natural number system is a set Z+ , an element 1 of Z+ , and a
function, s : Z+ → Z+ , called the successor operation with the
following 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
Peano Axioms
The natural number system is a set Z+ , an element 1 of Z+ , and a
function, s : Z+ → Z+ , called the successor operation with the
following 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+ .
Peano Axioms
The natural number system is a set Z+ , an element 1 of Z+ , and a
function, s : Z+ → Z+ , called the successor operation with the
following 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 for
every proof by “induction”
Theorem
Theorem
Suppose that M is a subset of the set Z+ of natural numbers with the
following two properties:
Theorem
Suppose that M is a subset of the set Z+ of natural numbers with the
following two properties:
1
The natural number 1 ∈ M.
Theorem
Suppose that M is a subset of the set Z+ of natural numbers with the
following two properties:
1
The natural number 1 ∈ M.
2
If the natural number k belongs to M, then the next natural number
k + 1 belongs to M.
Theorem
Suppose that M is a subset of the set Z+ of natural numbers with the
following two properties:
1
The natural number 1 ∈ M.
2
If the natural number k belongs to M, then the next natural number
k + 1 belongs to M.
Then all natural numbers belong to M, that is M = Z+ .
Template for a Proof by Induction
Template for a Proof by Induction
Theorem
A statement involving (positive) integers.
Template for a Proof by Induction
Theorem
Proof
A statement involving (positive) integers.
Our proof is by induction.
Template for a Proof by Induction
Theorem
Proof
A statement involving (positive) integers.
Our proof is by induction.
Basis:
We show that our statement holds for n = 1 (or another
suitable integer).
Template for a Proof by Induction
Theorem
Proof
A statement involving (positive) integers.
Our proof is by induction.
Basis:
We show that our statement holds for n = 1 (or another
suitable integer).
Induction hypothesis:
We assume that our statement holds for n = k.
Template for a Proof by Induction
Theorem
Proof
A statement involving (positive) integers.
Our proof is by induction.
Basis:
We show that our statement holds for n = 1 (or another
suitable integer).
Induction hypothesis:
We assume that our statement holds for n = k.
(For yourself, make sure you understand what the statement would say for
n = k + 1.)
Template for a Proof by Induction
Theorem
Proof
A statement involving (positive) integers.
Our proof is by induction.
Basis:
We show that our statement holds for n = 1 (or another
suitable integer).
Induction hypothesis:
We assume that our statement holds for n = k.
(For yourself, make sure you understand what the statement would say for
n = k + 1.)
Show that the statement holds for n = k + 1.
(You should use the induction hypothesis here.)
Template for a Proof by Induction
Theorem
Proof
A statement involving (positive) integers.
Our proof is by induction.
Basis:
We show that our statement holds for n = 1 (or another
suitable integer).
Induction hypothesis:
We assume that our statement holds for n = k.
(For yourself, make sure you understand what the statement would say for
n = 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 the
statement holds for all integers greater than or equal to 1.
Strong Induction
Strong Induction
Theorem
Another statement involving (positive) integers.
Strong Induction
Theorem
Proof
Another statement involving (positive) integers.
Our proof is by induction.
Strong Induction
Theorem
Proof
Another statement involving (positive) integers.
Our proof is by induction.
Basis:
We show that our statement holds for n = 1 (or another
suitable integer).
Strong Induction
Theorem
Proof
Another statement involving (positive) integers.
Our proof is by induction.
Basis:
We show that our statement holds for n = 1 (or another
suitable integer).
Induction hypothesis:
We assume that our statement holds for all
integers n = 1, 2, 3, . . . , k.
Strong Induction
Theorem
Proof
Another statement involving (positive) integers.
Our proof is by induction.
Basis:
We show that our statement holds for n = 1 (or another
suitable integer).
Induction hypothesis:
We assume that our statement holds for all
integers n = 1, 2, 3, . . . , k.
(For yourself, make sure you understand what the statement would say for
n = k + 1.)
Strong Induction
Theorem
Proof
Another statement involving (positive) integers.
Our proof is by induction.
Basis:
We show that our statement holds for n = 1 (or another
suitable integer).
Induction hypothesis:
We assume that our statement holds for all
integers n = 1, 2, 3, . . . , k.
(For yourself, make sure you understand what the statement would say for
n = k + 1.)
Show that the statement holds for n = k + 1.
(You should use the induction hypothesis here.)
Strong Induction
Theorem
Proof
Another statement involving (positive) integers.
Our proof is by induction.
Basis:
We show that our statement holds for n = 1 (or another
suitable integer).
Induction hypothesis:
We assume that our statement holds for all
integers n = 1, 2, 3, . . . , k.
(For yourself, make sure you understand what the statement would say for
n = 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 the
statement holds for all integers greater than or equal to 1.