Question:
Let \( \mathbb{Q}^+ \) be the set of positive rational numbers. If the binary operation \( \odot \) is defined by:
\[ a \odot b = \frac{ab}{2} \]
Find the inverse of 3.
Options:
- (a) \( \frac{4}{3} \)
- (b) 2
- (c) \( \frac{1}{3} \)
- (d) \( \frac{2}{3} \)
Solution:
Step 1: Find identity element
Let identity be \( e \), then:
\[ a \odot e = a \Rightarrow \frac{ae}{2} = a \]
\[ ae = 2a \Rightarrow e = 2 \]
Step 2: Find inverse of 3
Let inverse be \( x \), then:
\[ 3 \odot x = 2 \]
\[ \frac{3x}{2} = 2 \]
\[ 3x = 4 \Rightarrow x = \frac{4}{3} \]
Final Answer:
\[ \boxed{\frac{4}{3}} \]
Correct Option: (a)