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  • Let * be the binary operation defined on Q. Find which of the following binary operations are commutative (i) a * b = a – b ∀ a, b ∈Q

Let  \ast   be the binary operation defined on Q. Find which of the following binary operations are commutative

\\(i) a \ast b = a - b\: \: \forall a, b \in Q\\ (ii) a \ast b = a^2 + b^2 \: \: \forall a, b \in Q\\ (iii) a \ast b = a + ab \: \: \forall a, b \in Q\\ (iv) a \ast b = (a - b)\textsuperscript{2}\: \: \forall a, b \in Q\\

Answers (1)

Here,  \ast   is a binary operation defined on Q.

(i)

 \\a \ast b = a - b, \forall a, b \in Q \: \: and \: \: b \ast a = b - a\\ So, a \ast b \neq b \ast a\\

Hence, \ast    is not commutative.

(ii)

\\a \ast b = a\textsuperscript{2}+ b\textsuperscript{2}\\ b \ast a = b\textsuperscript{2}+ a\textsuperscript{2}\\

Therefore,  \ast   is commutative.

(iii)

\\a \ast b = a + ab\\ b \ast a = b + ab\\ Hence, a + ab \neq b + ab\\

Therefore,  \ast   is not commutative.

(iv)
\\a \ast b = (a - b)\textsuperscript{2}, \forall a, b \in Q\\ b \ast a = (b -a)\textsuperscript{2}\\ As, (a - b)\textsuperscript{2} = (b - a)\textsuperscript{2}\\

Therefore, \ast   is commutative.

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