Question

Let S be the set of all rational numbers of the form \[\frac{m}{n}\] , where m ∈ Z and n = 1, 2, 3. Prove that * on S defined by a * b = ab is not a binary operation.

Answer 1

\[S = \left\{ a = \frac{m}{n} : m \in Z, n \in \left\{ 1, 2, 3 \right\} \right\}\] 
Let \[a = \frac{1}{3}, b = \frac{5}{3} \in S\] 
\[a * b = ab \] 
\[ = \frac{1}{3} \times \frac{5}{3}\] 
\[ = \frac{5}{9} \not\in S \left[ \because 9 \not\in \left\{ 1, 2, 3 \right\} \right]\] 
\[\text{Therefore},\exists \text{ a, b} \ \text{ in S,such thata } * b\not\in S \]

Thus, * is not a binary operation.