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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 _______________ .
Options
closure
commutative
associative
none of these
Solution
none of these
* is not closure because when a = 1 and b = 2,
\[a * b = \frac{a}{b} = \frac{1}{2} \not\in Z\]
* is not commutative because when a = 1 and b = 2,
\[1 * 2 = \frac{1}{2}\]
\[2 * 1 = \frac{2}{1}\]
\[1 * 2 \neq 2 * 1\]
* is not associative because when a = 1, b = 2 and c = 3,
\[1 * \left( 2 * 3 \right) = 1 * \left( \frac{2}{3} \right)\]
\[ = \frac{1}{\left( \frac{2}{3} \right)}\]
\[ = \frac{3}{2}\]
\[\left( 1 * 2 \right) * 3 = \frac{1}{2} * 3\]
\[ = \frac{\left( \frac{1}{2} \right)}{3}\]
\[ = \frac{1}{6}\]
\[\text{Thus }, \]
\[1 * \left( 2 * 3 \right) \neq \left( 1 * 2 \right) * 3\]
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