Question

Determine whether the following operation define a binary operation on the given set or not :

\[' * ' \text{on Q defined by } a * b = \frac{a - 1}{b + 1} \text{for all a, b} \in Q .\]

Answer 1

If a = 2 and b = \[-\] 1 in Q,

\[a * b = \frac{a - 1}{b + 1}\] 
\[ = \frac{2 - 1}{- 1 + 1}\] 
\[ = \frac{1}{0} \left[ \text{ which is not defined } \right]\] 
\[\Rightarrow \text{ Fora} = 2 \text{ and }b = - 1, \] 
\[a * b \not\in Q\]

So, * is not a binary operation on Q.