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Question

If A = N x N and * be any binary operation on A defined by

(a, b) * (c, d) = (a + c, b + d), then the binary operation is


A

Associative

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B

Commutative

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C

Commutative and associative

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D

Commutative but not associative

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Solution

The correct option is C

Commutative and associative


(a, b) * (c, d) = (a + c, b + d) and

(c, d) * (a, b) = (c + a, d + b) = (a + c, b + d) = (a, b) * (c, d)

So, * is commutative.

(a, b) * [(c, d) * (e, f)] = (a, b) * (c + e, d + f) = (a + c + e, b + d + f) and

[(a, b) * (c, d)] * (e, f) = (a + c, b + d) * (e, f) = (a + c + e, b + d + f).

So, * is associative also.


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