Proof techniques

Question

Answer 1

• A. Using Direct Proof : Let’s assume n = 2x+1 and m = 2y+1 (using definitions of odd numbers). n+m = 2(x+y+1) => n+m = 2(some integer). Therefore n+m is even (using definitions of even numbers).
• B. Using Direct Proof : Let’s assume n = 2x+1 and m = 2y (using definitions of odd and even numbers). n+m = 2(x+y)+1 => n+m = 2(some integer)+1. Therefore n+m is odd.
• C. Using Direct Proof : Let’s assume n = 2x and m = 2y (using definitions of even numbers). n+m = 2(x+y) => n+m = 2(some integer). Therefore n+m is even.
• D. Using Direct Proof : Let’s assume n = 2x+1 and m = 2y+1 (using definitions of odd numbers). nm = (2x+1)(2y+1) = 2(2xy+x+y)+1 => nm = 2(some integer)+1. Therefore nm is odd.

(x,y are some integers)