Types of Proofs Types of Mathematical Statements 1

Types of Proofs

Types of Mathematical Statements • 1. Axiom/Postulate ……. . a statement that is assumed to be true without proof. • Example: There is one and only one straight line between two points. 2

2. Conjecture • a statement that is unproved, but is believed to be true • Example: There are infinite twin primes. • A twin prime is a prime number that has a prime gap of two. Eg. (11, 13), (5, 7), (29, 31), …. 3

3. Theorem • a mathematical statement that is proved using rigorous mathematical reasoning. 4

4. Lemma • a minor result whose sole purpose is to help in proving a theorem. • • It is a stepping stone on the path to proving a theorem. 5

5. Corollary • a direct consequence of theorem • a result in which the (usually short) proof relies heavily on a given theorem (we often say that “this is a corollary of Theorem A”). 6

Proof methods • We will discuss 6 proof methods: 1. Direct proofs 2. Indirect proofs 3. Proof by contradiction 4. Proof by cases 5. Counterexamples 6. Proof by PMI (principle of Mathematical induction) 7

1. Direct proofs • Consider an implication: p→q To perform a direct proof, assume that p is true, and show that q must therefore be true 8

Example Show that “the square of an even number is an even” Rephrased: If n is even, then n 2 is even. Proof: Assume n is even – Thus, n = 2 k, for some k (definition of even numbers) – n 2 = (2 k)2 = 4 k 2 = 2(2 k 2) – As n 2 is 2 times an integer, n 2 is thus even. – QED (Q. E. D. is an acronym for "quod erat demonstrandum", which is Latin for "which was to be demonstrated. ") 9

2. Indirect proofs (proof by Contrapositive) • Consider an implication: p→q – Its contrapositive is ¬q→¬p • Is logically equivalent to the original implication! – Thus, show that if ¬q is true, then ¬p is true • To perform an indirect proof, do a direct proof on the contrapositive 10

Example: Prove that “If n 2 is an odd integer, then n is an odd integer. ” • Prove the contrapositive: If n is an even integer, then n 2 is an even integer. • Proof: n=2 k for some integer k (definition of even numbers) • n 2 = (2 k)2 = 4 k 2 = 2(2 k 2) • Since n 2 is 2 times an integer, it is even 11

3. Proof by contradiction • To prove a statement is true – you may assume that it is false – And then proceed to show that such an assumption leads a contradiction with a known result Example: Prove that √ 2 is an irrational number 12

4. Proof by cases • Sometimes it is easier to prove a theorem by – Breaking it down into cases and – Proving each one separately 13

Example • Prove that –b≠ 0 • Cases: – Case 1: a ≥ 0 and b > 0 • Then |a| = a, |b| = b, and – Case 2: a ≥ 0 and b < 0 • Then |a| = a, |b| = -b, and – Case 3: a < 0 and b > 0 • Then |a| = -a, |b| = b, and – Case 4: a < 0 and b < 0 • Then |a| = -a, |b| = -b, and 14

The think about proof by cases • Make sure you get ALL the cases – The biggest mistake is to leave out some of the cases 15

5. Counter-examples • Given a universally quantified statement, find a single example which it is not true • Note that this is DISPROVING a UNIVERSAL statement by a counterexample Example. The square of any real number is positive. 16

Home work 1. 2. 3. Prove that if x 3 <0, then x<0 There is no greatest even integer. Let n ∈ Z. Prove that 9 n 2+3 n-2 is even 4. Disprove n 2+n+1 is a prime number for all n≥ 1