Question:
Let \( \mathbb{Q}^+ \) be the set of all positive rational numbers. The binary operation \( * \) is defined by:
\[ a * b = \frac{ab}{2} \]
Find the inverse of an element \( a \in \mathbb{Q}^+ \).
Options:
- (a) \( a \)
- (b) \( \frac{1}{a} \)
- (c) \( \frac{2}{a} \)
- (d) \( \frac{4}{a} \)
Solution:
Step 1: Find identity element
Let identity be \( e \), then:
\[ a * e = a \Rightarrow \frac{ae}{2} = a \]
\[ ae = 2a \Rightarrow e = 2 \]
Step 2: Find inverse of \( a \)
Let inverse be \( x \), then:
\[ a * x = 2 \]
\[ \frac{ax}{2} = 2 \]
\[ ax = 4 \Rightarrow x = \frac{4}{a} \]
Final Answer:
\[ \boxed{\frac{4}{a}} \]
Correct Option: (d)