Contrapositive Proof Exploration
Fill in the blank for the following proof:
Proposition: Suppose x ∈ Z. If x2 −6x+5 is even, then x is odd. Proof. (Contrapositive) Suppose x is (a)__________________________. Thus x is even, so (b)________________________ for some integer a. So x2 - 6x+5 =(2a)2−6(2a)+5 = 4a2−12a+5 = 4a2−12a+4+1 = 2(2a2−6a+2)+1. Therefore x2 −6x+5 = 2b +1, where b is the integer (d)________________________ Consequently x2 −6x+5 is (e)____________________. Therefore x2 −6x+5 is (f)________________________.
Your Turn:
Use the method of contrapositive proof to prove the following statement: Suppose x, y ∈ R. If y3 + yx2 ≤ x3 + x y2, then y ≤ x.