Prove that for ever two consecutive integers have opposite parity. Prove that

Question

Logic and proof techniques

Answer 1

If by parity you mean the quality of being an even number, then if one number, n, has even parity then n+1 has odd parity, because 1 is odd and even+odd=even. If n has odd parity then it can be represented by 2m+1 where m is an integer. 2m+1+1=2m+2=2(m+1) which is even. This is opposite to n which is odd. So any pair of consecutive integers have opposite parity.