State whether the following statement are true or false. Justify
(i) for an arbitrary binary operation $$ on a set N, $a a = a \forall a \in N$ .

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

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Solution

Define an operation $$ on N as $a b = a + b \forall a, b \in N$
Then, in particular, for $b = a = 3$ , we have $3 * 3 = 3 + 3 = 6 \neq 3$
Therefore, statement (i) is false.

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

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