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)= (a * b)* a

Answer 1

(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. **

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(ii) R.H.S. = (c * b) * a
= (b * c) * a [* is commutative]
= a * (b * c) [Again, as * is commutative]
= L.H.S. ∴ a * (b * c) = (c * b) * a
Therefore, statement (ii) is true