Answer
See below.
Work Step by Step
(a) To prove the statement by contraposition, we need to prove "for any real number $x$, if it is not an irrational, then its reciprocal $1/x$ will not be an irrational. To prove, let $x=\frac{a}{b}$ be a rational number where $a,b$ are non-zero integers, we have $\frac{1}{x}=\frac{b}{a}$ will also be a rational number (not an irrational). Thus we proved the statement by contraposition.
(b) To prove the statement by contradiction, suppose there exist an irrational $y$ whose reciprocal is not an irrational, let the reciprocal be
$\frac{1}{y}=\frac{c}{d}$ where $c,d$ are non-zero integers. We have
$y=\frac{d}{c}$ will also be a rational which contradicts to the condition that $y$ should be an irrational. Thus, we proved the statement by contradiction.
