Q. 9 D5.0( 4 Votes )

# Let ∗ be a binary operation on the set Q of rational numbers as follows:

a ∗ b = (a – b)^{2}

Find which of the binary operations are commutative and which are associative.

Answer :

It is given that ∗ be a binary operation on the set Q of rational numbers is defined as

a ∗ b = (a – b)^{2}

For a, b ϵ Q, we have,

a ∗ b = (a – b)^{2}

b ∗ a = (b – a)^{2} = [-(a – b)]^{2} = (a –b)^{2}

⇒ a * b = b * a

⇒ the operation * is commutative.

Also, We can see that (1 * 2) * 3 ≠ 1 *(2 * 3), where 1,2,3 ϵ Q

Therefore, the operation * is not associative.

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