Proof Methods Strategy Proof by Cases p 1

Proof Methods & Strategy

Proof by Cases • ( p 1 p 2 . . . pn ) q ( p 1 q ) ( p 2 q ) . . . ( pn q ). • Prove the compound implication by proving each implication. • Why do we have to prove all of them? [ x P( x ) y Q( y ) ] x y ( P( x ) Q( y ) [ x P( x ) x y ( P( x ) Q( y ) ] [ y Q( y ) ) x y ( P( x ) Q( y ) ] Copyright © Peter Cappello 2

Example • Let the domain be the set of integers. • a ( ( 3 | a – 1 3 | a – 2 ) 3 | a 2 – 1 ). Proof: WHAT WHY 1. Let a be an arbitrary integer. 2. Case 3 | a - 1: 1. Let integer q = ( a – 1 ) / 3. 2. a = 3 q + 1. 3. a 2 = ( 3 q + 1 )2 = 9 q 2 + 6 q + 1 = 3( 3 q 2 + 2 q ) + 1 4. a 2 – 1 = 3(3 q 2 + 2 q ). 5. (3 divides a – 1 with no remainder). 3 | a 2 – 1 3. Case 3 | a - 2: (Proved previously). 4. a ( ( 3 | a – 1 3 | a – 2 ) 3 | a 2 – 1 ). (Universal Generalization) Copyright © Peter Cappello 3

Existence Proofs x P( x ) • Constructive Produce a particular x such that P( x ) is true. • Non-constructive Prove that there is an x such that P( x ) is true without actually producing it. Copyright © Peter Cappello 4

A Constructive Existence Proof A constructive proof that constructive proofs exist • Let the domain be natural numbers • Prove: There are distinct natural numbers a, b, c, d such that a 3 + b 3 = c 3 + d 3. Proof: 103 + 93 = 1000 + 729 = 1728 + 1 = 123 + 13. Copyright © Peter Cappello 5

A Non-Constructive Existence Proof If 1, 2, …, 10 are placed randomly in a

Uniqueness Proof • There is exactly 1 x that makes P( x ) true. • x ( P( x) y ( x y P( y ) ) ). • Alternatively (contrapositive of 2 nd part), x ( P( x ) y ( P( y ) y = x ) ). Copyright © Peter Cappello 7

Example • Domain: Integers • Prove: There is a unique additive identity: x a ( a + x = a y ( a + y = a x = y ) ). Proof: 1. Let a be an arbitrary integer. 2. a+0=a 3. Let y be an arbitrary integer. 4. Assume a + y = a. 5. a + 0 = a + y 0 = y. (From steps 2 & 4. ) 6. x a ( a + x = a y ( a + y = a y = x ) ). (There is an additive identity for integers: x a a + x = a ) Copyright © Peter Cappello 8