Question

Let * be a binary operation defined on Q. Which of the following binary operation(s) is/are associative?

• A. a * b = a – b for a, b ∈ Q.
• B. a * b = $\frac{ab}{4}$ for a, b ∈ Q.
• C. a * b = a – b + ab for a, b ∈ Q
• D. a * b = $a{b}^2$ for a, b ∈ Q

Answer 1

The correct option is B a * b = $\frac{ab}{4}$ for a, b ∈ Q.

(i) * is not associative for if we take
a = 1, b = 2 and c = 3, then
(a * b) * c = (1 * 2) * 3
= (1 – 2) * 3
= – 1 – 3 = – 4,

and a * (b * c) = 1 * (2 * 3)
= 1 * (2 – 3)
= 1 – ( – 1) = 2.
Thus (a * b) * c ≠ a * (b * c) and hence
* is not associative.

(ii) * is associative since Q is associative with respect to multiplication.

(iii) * is not associative for if we take
a = 2, b = 3 and c = 4, then
(a * b) * c = (2 * 3) * 4
= (2 – 3 + 6) * 4
= 5 * 4
= 5 – 4 + 20 = 21,

and a * (b * c) = 2 * (3 * 4)
= 2 * (3 – 4 + 12)
= 2 * 11 = 2 – 11 + 22 = 13
Thus (a * b) * c ≠ a * (b * c) and hence * is not associative.

(iv) * is not associative for if we take
a = 1, b = 2 and c = 3, then
(a * b) * c = (1 * 2) * 3
= 4 * 3
= 4 × 9 = 36,

and a * (b * c) = 1 * (2 * 3)
= 1 * 18 = 1 × 18^2 = 324.
Thus (a * b) * c ≠ a * (b * c) and hence * is not associative.