Question:
On the set \( \mathbb{Z} \) of all integers, a binary operation \( * \) is defined by:
\[ a * b = a + b + 2 \]
Find the inverse of 4.
Solution:
Step 1: Find Identity Element
Let identity be \( e \), then:
\[ a * e = a \]
\[ a + e + 2 = a \]
\[ e + 2 = 0 \Rightarrow e = -2 \]
Step 2: Find inverse of 4
Let inverse of 4 be \( x \), then:
\[ 4 * x = e = -2 \]
\[ 4 + x + 2 = -2 \]
\[ x + 6 = -2 \]
\[ x = -8 \]
Step 3: Verify
\[ 4 * (-8) = 4 – 8 + 2 = -2 \]
Which equals identity, so correct.
Final Answer:
\[ \boxed{-8} \]