Question:
Let \( * \) be defined on non-zero integers by:
\[ a * b = \frac{a}{b} \]
Which property is satisfied?
- (a) Closure
- (b) Commutative
- (c) Associative
- (d) None of these
Solution:
1. Closure:
For closure, result must be an integer. Example:
\[ 1 * 2 = \frac{1}{2} \notin \mathbb{Z} \]
So, not closed ❌
—2. Commutativity:
\[ a * b = \frac{a}{b}, \quad b * a = \frac{b}{a} \]
Clearly:
\[ \frac{a}{b} \neq \frac{b}{a} \]
So, not commutative ❌
—3. Associativity:
\[ (a * b) * c = \left(\frac{a}{b}\right) * c = \frac{a}{bc} \]
\[ a * (b * c) = a * \left(\frac{b}{c}\right) = \frac{ac}{b} \]
Since:
\[ \frac{a}{bc} \neq \frac{ac}{b} \]
So, not associative ❌
—Final Answer:
\[ \boxed{\text{(d) None of these}} \]