Question:
Let \( * \) be a binary operation on \( \mathbb{Q} – \{1\} \) defined by:
\[ a * b = a + b – ab \]
Find the identity element.
Options:
- (a) 1
- (b) \( \frac{a-1}{a} \)
- (c) \( \frac{a}{a-1} \)
- (d) 0
Solution:
Step 1: Let identity be \( e \), then
\[ a * e = a \]
\[ a + e – ae = a \]
Step 2: Simplify
\[ e – ae = 0 \Rightarrow e(1 – a) = 0 \]
Since \( a \neq 1 \), we must have:
\[ e = 0 \]
Step 3: Verify
\[ a * 0 = a + 0 – 0 = a \]
Verified.
Final Answer:
\[ \boxed{0} \]
Correct Option: (d)