Let ∗ be a binary operation on the set Q of rational numbers as follows:
a ∗ b = (a – b)2
Find which of the binary operations are commutative and which are associative.
It is given that ∗ be a binary operation on the set Q of rational numbers is defined as
a ∗ b = (a – b)2
For a, b ϵ Q, we have,
a ∗ b = (a – b)2
b ∗ a = (b – a)2 = [-(a – b)]2 = (a –b)2
⇒ a * b = b * a
⇒ the operation * is commutative.
Also, We can see that (1 * 2) * 3 ≠ 1 *(2 * 3), where 1,2,3 ϵ Q
Therefore, the operation * is not associative.