Question:
Let \( * \) be defined on \( \mathbb{Q} – \{1\} \) by:
\[ a * b = a + b – ab \]
Find the identity element.
Options:
- (a) 0
- (b) 1
- (c) \( \frac{1}{2} \)
- (d) -1
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
\[ a * 0 = a + 0 – 0 = a \]
So, identity exists and is valid in the set.
—Final Answer:
\[ \boxed{0} \]
Correct Option: (a)