Question:
Which of the following statements is true?
- A. \( a * b = \frac{a+b}{2} \) is a binary operation on \( \mathbb{Z} \)
- B. \( a * b = \frac{a+b}{2} \) is a binary operation on \( \mathbb{Q} \)
- C. All commutative operations are associative
- D. Subtraction is a binary operation on \( \mathbb{N} \)
Solution:
Option A:
\[ \frac{a+b}{2} \] is not always an integer (e.g., \(1,2 \Rightarrow 3/2\)). So, not closed in \( \mathbb{Z} \) ❌
Option B:
Sum of rationals is rational and division by 2 keeps it rational. So, closed in \( \mathbb{Q} \) ✅
Option C:
Commutative does not imply associative (counterexample exists). ❌
Option D:
Subtraction is not closed in \( \mathbb{N} \) (e.g., \(2 – 5 = -3\)). ❌
Final Answer:
\[ \boxed{\text{B}} \]