Question

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

a * b = a – b + ab for a, b ∈ Q

Answer 1

* 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.