Show that the following statement is true by the method of contra… - Vidyadip

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.

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Answer

$p$ : 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 contra positive 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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