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Section 2.3. Mathematical induction
Consider the sequence: 1, 3, 9, 27, 81, …
recurrence relation:
Consider the summation: 1 + 3 + 9 + …
Let’s look at the sums for the first few values of n
n
1 + 3 + … + 3n
0
1
1
2
3
4
n
Definition: A statement about the positive integers is a predicate
P(n) with the set of positive integers as its domain. That is, when
any positive integer is substituted for n in the statement P(n) the
result is a proposition that is unambiguously either true or false.
Induction Proofs
n
 (2i – 1) = 1 + 3 + 5 + 7 + … + 2n - 1
i=1
Show that n  4 cents of postage can be obtained using only 2
cent and 5 cent stamps.
n
4 cents
5 cents
6 cents
7 cents
8 cents
9 cents
10 cents
The basic approach for a mathematical induction proof:
How to do an Induction Proof
n
Prove that  (2i – 1) = 1 + 3 + 5 + 7 + … + 2n – 1 = n2
i=1
(1) Basis:
(2) State the hypothesis: Assume
(3) Induction step
Another example
n
 i = n(n + 1)/2
i=1
Basis:
Hypothesis: Assume
Induction step:
Example: You have invested $500 at 6% interest compounded
annually. If you make no withdrawals, prove that after n years the
account contains (1.06)n•500 dollars.
Basis:
Hypothesis: Assume
Induction step:
Yet Another Example
Prove that 1•2 + 2•3 + ... +n(n + 1) = n(n + 1)(n + 2)/3
Example of a product
Prove that (1 – 1/22)•(1 – 1/32)• … •(1 – 1/n2) = (n + 1)/2n
Basis:
Hypothesis: Assume
Induction step:
Proving an inequality
Example: For all integers n ≥ 3, 2n + 1 < 2n
Basis:
Hypothesis: Assume
Induction step:
Example: Prove n2 < 2n whenever n is an integer greater than 4.
Basis:
Hypothesis: Assume
Induction Step:
More Inequality Examples
Example 1: Prove that (1 + ½)n ≥ 1 + n/2 for n ≥ 0.
Basis:
Hypothesis: Assume (1 + ½)k ≥ 1 + k/2 for k ≥ 0.
Induction step:
Example 2: Prove that 2n < n! for n ≥ 4
Basis: n = 4
Hypothesis: Assume 2k < k! for k ≥ 4
Induction step:
More Examples
Example 3: Use mathematical induction to prove that a set with n
elements has n(n - 1)/2 subsets of size 2.
Basis: n = 2.
Hypothesis: Assume if S is a set and |S| = k, then S has k(k-1)/2
subsets of size 2.
Induction step:
Case 1:
Case 2:
Still More Examples
Example 4: Prove that 5 | (n5 - n) whenever n is a nonnegative
integer.
Basis:
Hypothesis: Assume 5 | (k5 - k) for k ≥ 0
Induction step:
Proving this directly:
Strong vs weak induction
Up until now we have proceeded as follows:
1.
2.
3.
Strong Induction:
1.
2.
3.
Fundamental theorem of arithmetic:
Theorem 1 (p. 129) Every integer greater than 1 can be expressed
as the product of a list of prime numbers.
Another Strong Induction Example
Consider compound propositions using only the connectives , V
and  where propositions are parenthesized before being joined.
Count each letter, connective or parenthesis as one symbol. For
example, (p)  (q) has 7 symbols.
Prove any compound proposition constructed in this way has an
odd number of symbols.
Basis case:
Hypothesis: Assume that if a compound proposition formed in
this way has r symbols, 1 ≤ r ≤ k, then r is odd.
Induction step:
Section 2.5 Contradiction and the Pigeonhole Principle
In a direct proof we assume
and prove
In a contrapositive proof we assume
In a proof by contradiction we assume
and prove
and
Read propositions 1 – 3.
Theorem 4: The real number 2 is irrational. That is, there does
not exist a rational number r such that r2 = 2.
Definition: Two numbers are relatively prime if they have no common divisor greater than 1.
Proof that 2 is irrational
Constructive and Nonconstructive Proofs
Proving a statement of the type x P(x)
constructive: Find a specific value of x that makes the statement true.
Example: For every integer n there is an even integer x such that x > n
In a nonconstructive proof
Example: For every positive integer n there is a prime number p
such that p > n
Difference between contradiction and contrapositive proofs
Prove that if n is an integer and n3 + 5 is odd, then n is even.
Contrapositive Proof: Suppose n is odd.
Proof by contradiction:
Another example: If a number added to itself gives itself, then the
number is 0.
Frequently, a direct or contrapositive proof is hiding in a supposed proof by contradiction.
The Pigeonhole Principle
Pigeonhole principle:
Example 1: In any set of five numbers, there are two numbers x
and y that have the same remainder when divided by 4.
Example 2: If five numbers are chosen from the set {1, 2, 3, 4, 5,
6, 7, 8}, then two of the numbers chosen must have a sum of 9.
The generalized pigeonhole principle:
Example 3:
Example 4:
Example 5:
More Examples
Example 6: Given a group of six acquaintances prove there are 3
mutual friends or 3 mutual enemies.
Example 7: How many students each of whom comes from one of
the 50 states must be enrolled at a university to guarantee there
are at least 100 who come from the same state?
Example 8: How many people must be in a room to guarantee
that two people have the same first and last initials?