Question

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

a * b = ab² 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 = 4 * 3 = 4 × 9 = 36

And a * (b * c) = 1 * (2 * 3) = 1 * 18 = 1 × 182 = 324.

Thus (a * b) * c ≠ a * (b * c) and hence * is not associative.