Question

State whether the following statements are true or false. Justify. (i) For an arbitrary binary operation * on a set N, a * a = a ∀ a * N. (ii) If * is a commutative binary operation on N, then a * (b * c) = (c * b) * a

Answer 1

(i) For an arbitrary binary operation ∗ on a set N, a∗ a = a ∀a ∈ N.
The above statement is false.
$\text{\underline{explanation}}$: It is given that an operation ∗ on a set N, a ∗ a = a ∀a ∈ N
if a* b = a + b where a , b ∈N .
then, a * a = a + a = 2a ≠ a.
for example, b = a = 3, we get,
3 * 3 = 3 + 3 = 6 ≠ 3

(ii) If ∗ is a commutative binary operation on N, then a ∗ (b ∗ c) = (c ∗ b) ∗ a
The above statement if true.
$\text{\underline{Explanation}}$: RHS = (c * b) * a
= (b * c) * a (* is commutative)
= a * (b * c) (as * is commutative)
= LHS.
Therefore, a * (b * c) = (c * b) * a.