Question

State whether the following statements are true or false. Justify.

(i) For any arbitrary binary operation * on a set N,

a * a = a ∀a ∈ N.

(ii) If * is commutative binary operation on N, then a * (b * c) = (c * b) * a.

Answer 1

(i) A binary operation on N is defined as
a * a = a ∀a ∈ N.
Here, operation * is not defined.
∴ Given statement is false.

(ii) * is a binary commutative operation on N.
⇒ c * b = b * c [∵ * is commutative]
∴ (c * b) * a = (b * c) * a = a * (b * c)
∴ a * (b * c) = (c * b) * a.
∴ This statement is true.