Question:
For the binary operation \( * \) on \( \mathbb{Z} \) defined by:
\[ a * b = a + b + 1 \]
Find the identity element.
Options:
- (a) 0
- (b) -1
- (c) 1
- (d) 2
Solution:
Step 1: Let identity be \( e \), then
\[ a * e = a \]
\[ a + e + 1 = a \]
Step 2: Solve for \( e \)
\[ e + 1 = 0 \Rightarrow e = -1 \]
Step 3: Verify
\[ a * (-1) = a – 1 + 1 = a \]
Verified.
Final Answer:
\[ \boxed{-1} \]
Correct Option: (b)