Question:
Let \( * \) be defined on \( \mathbb{N} \) by:
\[ a * b = a + b + 10 \]
Find the identity element.
Options:
- (a) -10
- (b) 0
- (c) 10
- (d) Non-existent
Solution:
Step 1: Let identity be \( e \), then
\[ a * e = a \]
\[ a + e + 10 = a \]
Step 2: Solve
\[ e + 10 = 0 \Rightarrow e = -10 \]
Step 3: Check domain
But \( -10 \notin \mathbb{N} \).
So, identity does not belong to the set.
—Final Answer:
\[ \boxed{\text{(d) Non-existent}} \]