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
(i) Define an operation * on N as:
a * b = a + b a, b ∈ N
Then, in particular, for b = a = 3, we have:
3 * 3 = 3 + 3 = 6 ≠ 3
Therefore, statement (i) is false.
(ii) R.H.S. = (c * b) * a
= (b * c) * a [* is commutative]
= a * (b * c) [Again, as * is commutative]
∴ a * (b * c) = (c * b) * a
Therefore, statement (ii) is true.