Question

Let p : If x is an integer and x² is even, then x is even.

Using the method of contrapositive, prove that p is true.

Answer 1

Let p: x is an integer and x² is even.

q: x is even

For contrapositive,

~p = x is an integer and x² is not even.

~q = x is not even.

Now, the statement is: If x is an integer and x² is not even, then x is not even.

Proof:

Let x be an odd integer and x = 2n + 1

⇒x² = (2n + 1) ² = 4n² + 4n + 1 (odd integer)

Thus, if x is an integer and x² is not even, then x is not even.