Prove that every integer is either even or odd, but not both. Solution Let us prove this by contradiction. Let us assume that a number can be both even and odd. Let the number be x. IF x is an even number, then x can be represented as : x = 2n IF x is an odd number, then it can be represented as some : x = 2m + 1 Equating the two : 2n = 2m + 1 2(n - m) = 1 In the above equation 2(n -m ) is an EVEN value because it is multiplied by 2 But it is equated to an ODD value, i.e 1 Since the above statement is FALSE, we can state that the function cannot be both even and odd. Hence proved that the function is either EVEN or ODD but not BOTH.