Question

An operation * is defined on the set Z of non-zero integers by $a*b=\frac{a}{b}$ for all a, b ∈ Z. Then the property satisfied is
(a) closure
(b) commutative
(c) associative
(d) none of these

Answer 1

(d) none of these

* is not closure because when a = 1 and b = 2,
$a*b=\frac{a}{b}=\frac{1}{2}\notin Z$

* is not commutative because when a = 1 and b = 2,
$$\begin{gathered} 1*2=\frac{1}{2} \\ 2*1=\frac{2}{1} \\ 1*2\ne 2*1 \end{gathered}$$

* is not associative because when a = 1, b = 2 and c = 3,
$$\begin{gathered} 1*\left(2*3\right)=1*\left(\frac{2}{3}\right) \\ =\frac{1}{\left({\displaystyle \frac{2}{3}}\right)} \\ =\frac{3}{2} \\ \left(1*2\right)*3=\frac{1}{2}*3 \\ =\frac{\left({\displaystyle \frac{1}{2}}\right)}{3} \\ =\frac{1}{6} \\ \mathrm{Thus}, \\ 1*\left(2*3\right)\ne \left(1*2\right)*3 \end{gathered}$$