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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?

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