State whether the following statement are true or false. Justify
(ii) If $$ is a commutative binary operation on N, then $a (b * c) = (c * b) * a .$

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Solution

$R H S = (c * b) * a = (b * c) * a [$ 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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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 $a * (b * c) = (c * b) * a .$

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

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N, a * a = a \forall a ϵ N

*

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a * b = \frac{a + b}{2}

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