Question

Let A be any set containing more than one element. Let ‘*’ be a binary operation on A defined by a * b = b for all a, b ∈ A Is ‘*’ commutative or associative on A?

Answer 1

Let a, b ∈ A

Then, a * b = b

b * a = a

So, a * b ≠ b * a

Thus, * is not commutative on A

Let us check associativity:

Let a, b, c ∈ A

a * (b * c) = a * c

= c

So

a * (b * c) = (a * b) * c, ∀ a, b, c ∈ A

Hence, * is associative on A