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Show that the following statement is true by the method of contrapositive.

Question:

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.

Solution:

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

Let $q: x$ is an integer and $x^{2}$ is even.

$r . x$ is even.

To prove that p is true by contrapositive method, we assume that r is false, and prove that q is also false.

Let x is not even.

To prove that $q$ is false, it has to be proved that $x$ is not an integer or $x^{2}$ is not even.

$x$ is not even implies that $x^{2}$ is also not even.

Therefore, statement q is false.

Thus, the given statement p is true.

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