📺 Watch Video Explanation:
Determine whether the operation is a binary operation or not
Given:
\( S = \left\{ \frac{m}{n} : m \in \mathbb{Z}, \; n = 1,2,3 \right\} \)
Operation:
\( a * b = ab \)
Concept:
A binary operation must satisfy closure.
Solution:
Take two elements from the set:
\( a = \frac{1}{2}, \quad b = \frac{1}{3} \)
Then:
\( a * b = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \)
But denominator 6 is not allowed (only 1, 2, 3 are allowed).
\( \frac{1}{6} \notin S \)
Conclusion:
The set is not closed under multiplication.
❌ Therefore, the operation is NOT a binary operation on \( S \).