Question

1. Consider the following claim. Claim: For two integers a and b, if a + b is odd then a is odd or b is odd.

(a) If we consider the claim as the implication P =⇒ Q, which statement is P and which is Q?

(b) Write the negations ¬P and ¬Q. (c) (1 point) Write the contrapositive of the claim.

(d) Prove the contrapositive of the claim.

2. Use contraposition (proof by contrapositive )to prove the following claim. Claim: If n is a positive integer such that n 2 > 25, then n > 5.

3. Use proof by contradiction to prove the following claim: Claim: There are not two integers a and b such that 88a + 44b = 1

Answer 1

1. given a+b is odd

a+b =2k +1 where k is some integer

It imply is is odd only when one of them is even and ither is odd

let a is odd and b is even

a=2m+1

b=2n

a+b=2n+2m+1=2(m+n)+1=2k+1.

Hence a or b is odd