Discrete Mathematics with Applications
MATH236
Dr. Hung P. Tong-Viet
School of Mathematics, Statistics and Computer Science
University of KwaZulu-Natal
Pietermaritzburg Campus
Semester 1, 2013
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Table of contents
1
Mathematical Induction
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Mathematical Induction
Principle of Mathematical Induction
Theorem
To prove that P(n) is true for all positive integers n, where P(n) is a
statement involving n, we need to complete two steps:
1
Basis Step: Verity that P(1) is true
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Mathematical Induction
Principle of Mathematical Induction
Theorem
To prove that P(n) is true for all positive integers n, where P(n) is a
statement involving n, we need to complete two steps:
1
Basis Step: Verity that P(1) is true
2
Inductive Step: Show that if P(k) is true, then P(k + 1) is true for all
integers k ≥ 1.
Tong-Viet (UKZN)
MATH236
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Mathematical Induction
Principle of Mathematical Induction
Theorem
To prove that P(n) is true for all positive integers n, where P(n) is a
statement involving n, we need to complete two steps:
1
Basis Step: Verity that P(1) is true
2
Inductive Step: Show that if P(k) is true, then P(k + 1) is true for all
integers k ≥ 1.
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Mathematical Induction
Example
Example
Use mathematical induction to show that for all nonnegative integer n,
1 + 2 + · · · + 2n = 2n+1 − 1
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Mathematical Induction
Example
Proof.
Let P(n) be the statement that 1 + 2 + · · · + 2n = 2n+1 − 1 for the
integer n.
Basis Step: P(0) is true because 20 = 21 − 1
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Mathematical Induction
Example
Proof.
Let P(n) be the statement that 1 + 2 + · · · + 2n = 2n+1 − 1 for the
integer n.
Basis Step: P(0) is true because 20 = 21 − 1
Inductive Step: Suppose that P(k) is true for an arbitrary
nonnegative integer k. That is, we assume that
1 + 2 + · · · + 2k = 2k+1 − 1
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Mathematical Induction
Example
Proof.
Let P(n) be the statement that 1 + 2 + · · · + 2n = 2n+1 − 1 for the
integer n.
Basis Step: P(0) is true because 20 = 21 − 1
Inductive Step: Suppose that P(k) is true for an arbitrary
nonnegative integer k. That is, we assume that
1 + 2 + · · · + 2k = 2k+1 − 1
We must show that P(k + 1) is true, that is, we must show that
1 + 2 + · · · + 2k+1 = 2k+2 − 1
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Mathematical Induction
Example
Proof.
Let P(n) be the statement that 1 + 2 + · · · + 2n = 2n+1 − 1 for the
integer n.
Basis Step: P(0) is true because 20 = 21 − 1
Inductive Step: Suppose that P(k) is true for an arbitrary
nonnegative integer k. That is, we assume that
1 + 2 + · · · + 2k = 2k+1 − 1
We must show that P(k + 1) is true, that is, we must show that
1 + 2 + · · · + 2k+1 = 2k+2 − 1
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Mathematical Induction
Example
We have that
1 + 2 + · · · + 2k + 2k+1
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Mathematical Induction
Example
We have that
1 + 2 + · · · + 2k + 2k+1
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= (1 + 2 + · · · + 2k ) + 2k+1
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Mathematical Induction
Example
We have that
1 + 2 + · · · + 2k + 2k+1
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= (1 + 2 + · · · + 2k ) + 2k+1
= (2k+1 − 1) + 2k+1
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Mathematical Induction
Example
We have that
1 + 2 + · · · + 2k + 2k+1
= (1 + 2 + · · · + 2k ) + 2k+1
= (2k+1 − 1) + 2k+1
= 2 · 2k+1 − 1 = 2k+2 − 1
By mathematical induction, we know that P(n) is true for all integers
n ≥ 0.
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Mathematical Induction
Example
We have that
1 + 2 + · · · + 2k + 2k+1
= (1 + 2 + · · · + 2k ) + 2k+1
= (2k+1 − 1) + 2k+1
= 2 · 2k+1 − 1 = 2k+2 − 1
By mathematical induction, we know that P(n) is true for all integers
n ≥ 0.
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Mathematical Induction
More examples
Example
1
Prove that 7n+2 + 82n+1 is divisible by 57 for every integers n ≥ 0.
2
Prove that n! > 2n for all integers n ≥ 4
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Mathematical Induction
More examples
Example
1
Prove that 7n+2 + 82n+1 is divisible by 57 for every integers n ≥ 0.
2
Prove that n! > 2n for all integers n ≥ 4
3
Prove that 1 · 1! + 2 · 2! + · · · + n · n! = (n + 1)! − 1 for every integer
n ≥ 1.
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Mathematical Induction
More examples
Example
1
Prove that 7n+2 + 82n+1 is divisible by 57 for every integers n ≥ 0.
2
Prove that n! > 2n for all integers n ≥ 4
3
Prove that 1 · 1! + 2 · 2! + · · · + n · n! = (n + 1)! − 1 for every integer
n ≥ 1.
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Mathematical Induction
Strong Induction
Theorem
To prove that P(n) is true for all positive integers n, where P(n) is a
statement involving n, we need to complete two steps:
1
Basis Step: Verify that P(1) is true
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Mathematical Induction
Strong Induction
Theorem
To prove that P(n) is true for all positive integers n, where P(n) is a
statement involving n, we need to complete two steps:
1
Basis Step: Verify that P(1) is true
2
Inductive Step: For all positive integer k ≥ 1, if P(1), P(2), · · · P(k)
is true, then P(k + 1) is true.
Tong-Viet (UKZN)
MATH236
Semester 1, 2013
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Mathematical Induction
Strong Induction
Theorem
To prove that P(n) is true for all positive integers n, where P(n) is a
statement involving n, we need to complete two steps:
1
Basis Step: Verify that P(1) is true
2
Inductive Step: For all positive integer k ≥ 1, if P(1), P(2), · · · P(k)
is true, then P(k + 1) is true.
Tong-Viet (UKZN)
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