Methods of Proof Lecture, Lecture notes of Discrete Mathematics

Download Methods of Proof Lecture and more Lecture notes Discrete Mathematics in PDF only on Docsity!

Methods of Proof

Lecture 3: Sep 9

This Lecture Now we have learnt the basics in logic. We are going to apply the logical rules in proving mathematical theorems.

• Direct proof
• Contrapositive
• Proof by contradiction
• Proof by cases

Proving an Implication Goal: If P, then Q. (P implies Q) Method 1: Write assume P, then show that Q logically follows. The sum of two even numbers is even. x = 2m, y = 2n x+y = 2m+2n = 2(m+n) Proof

Direct Proofs The product of two odd numbers is odd. x = 2m+1, y = 2n+ xy = (2m+1)(2n+1) = 4mn + 2m + 2n + 1 = 2(2mn+m+n) + 1. Proof If m and n are perfect square, then m+n+2√(mn) is a perfect square. Proof m = a 2 and n = b 2 for some integers a and b Then m + n + 2√(mn) = a 2

• b 2
• 2ab = (a + b)^2 So m + n + 2√(mn) is a perfect square.

Proving an Implication Claim: (^) If r is irrational, then √r is irrational. How to begin with? What if I prove “If √r is rational, then r is rational”, is it equivalent? Yes, this is equivalent, because it is the contrapositive of the statement, so proving “if P, then Q” is equivalent to proving “if not Q, then not P”. Goal: If P, then Q. (P implies Q) Method 1: Write assume P, then show that Q logically follows.

Rational Number R is rational  there are integers a and b such that and b ≠ 0. numerator denominator Is 0.281 a rational number? Is 0 a rational number? If m and n are non-zero integers, is (m+n)/mn a rational number? Is the sum of two rational numbers a rational number? Is x=0.12121212…… a rational number? Yes, 281/ Yes, 0/ Yes Yes, a/b+c/d=(ad+bc)/bd Note that 100x-x=12, and so x=12/99.

Proving an “if and only if” Goal: Prove that two statements P and Q are “ logically equivalent” , that is, one holds if and only if the other holds. Example: For an integer n, n is even if and only if n 2 is even. Method 1a: Prove P implies Q and Q implies P. Method 1b: Prove P implies Q and not P implies not Q. Method 2: Construct a chain of if and only if statement.

Proof the Contrapositive Statement: If n 2 is even, then n is even Statement: If n is even, then n 2 is even n = 2k n 2 = 4k 2 Proof: Proof: n 2 = 2k n = √(2k) ?? For an integer n, n is even if and only if n 2 is even. Method 1a: Prove P implies Q and Q implies P.

This Lecture

• Direct proof
• Contrapositive
• Proof by contradiction
• Proof by cases

P F

P

Proof by Contradiction

To prove P, you prove that not P would lead to ridiculous result, and so P must be true.

so can assume m  2 l 2 2 m  4 l 2 2 n  2 l so n is even. n m 2  2 n  m 2 2 2 n  m so m is even. 2 2 2 n  4 l Proof by Contradiction Theorem: (^) 2 is irrational. Proof (by contradiction): Want to prove both m and n are even. Recall that m is even if and only if m 2 is even.

Infinitude of the Primes Theorem. There are infinitely many prime numbers. Assume there are only finitely many primes. Let p 1 , p 2 , …, pN be all the primes. (1) We will construct a number N so that N is not divisible by any pi. By our assumption, it means that N is not divisible by any prime number. (2) On the other hand, we show that any number must be divided by some prime. It leads to a contradiction, and therefore the assumption must be false. So there must be infinitely many primes. Proof (by contradiction):

Infinitude of the Primes Theorem. There are infinitely many prime numbers. Claim : if p divides a, then p does not divide a+1. Let p 1 , p 2 , …, pN be all the primes. Consider p 1 p 2 …pN + 1. Proof (by contradiction): Proof (by contradiction): a = cp for some integer c a+1 = dp for some integer d => 1 = (d-c)p, contradiction because p>=2. So, by the claim, none of p 1 , p 2 , …, pN can divide p 1 p 2 …pN + 1, a contradiction.

This Lecture

• Direct proof
• Contrapositive
• Proof by contradiction
• Proof by cases