Q.Show that the following statement is true by the method of contrapositive.
p: If x is an integer and x$^2$ is even, then x is also even.

Statement p: "If x is an integer and $x^2$ is even, then x is also even."

Contrapositive of p: "If x is not even (i.e., x is odd), then $x^2$ is not even (i.e., $x^2$ is odd)."

Proof of the contrapositive: Suppose x is odd. Then $x = 2k+1$ for some integer k.

Since $2k^2+2k$ is an integer, $x^2$ is of the form $2(\text{integer})+1$, which is odd. …

State the contrapositive of "x² even ⇒ x even", which is "x odd ⇒ x² odd". Prove this directly by writing x = 2k+1 and expanding; the contra …

• Trying to prove the statement directly from x² even, which requires taking square roots and is much messier than proving the contrapositive. …