Question

Let * be a binary operation defined on Q. Find which of the following binary operations are associative

a * b = a – b for a, b ∈ Q

Answer 1

* 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.