Let * be binary operation defined on R by $p * q = \frac{p + q}{2}, \forall$ p, q ∈ R. The operation is:

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Question

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AAI ATC Junior Executive 21 Feb 2023 Shift 2 Official Paper

Answer 1

Given:

Let * be binary operation defined on R by $p * q = \frac{p + q}{2}, \forall$ p, q ∈ R

Concept:

If $p * q = q * p$ then relation is commutative.

If $(p * q) * r = p * (q * r)$ then relation is associative .

Calculation:

Let * be binary operation defined on R by $p * q = \frac{p + q}{2}, \forall$ p, q ∈ R

Commutative:

Let p, q ∈ R

Now,

$p * q = \frac{p + q}{2} = \frac{q + p}{2} = q * p$

Hence R is commutative.

Associative :

Let p, q, r ∈ R then

$(p * q) * r = \frac{p + q}{2} * r = \frac{p + q + 2 r}{4}$

$p * (q * r) = p * \frac{q + r}{2} = \frac{2 p + q + r}{4}$

then $(p * q) * r \neq p * (q * r)$

Hence R is not associative.

Hence the option (1) correct .

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