(A) The binary operation is defined as $a * b = a^{2} + b^{2}$.To check for commutativity,we need to verify if $a * b = b * a$ for all $a, b \in Q$.Gi

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(A) The binary operation is defined as $a * b = a^{2} + b^{2}$.To check for commutativity,we need to verify if $a * b = b * a$ for all $a, b \in Q$.Given $a * b = a^{2} + b^{2}$.Since addition is commutative in the set of rational numbers,we have $a^{2} + b^{2} = b^{2} + a^{2}$.By the definition of the operation,$b^{2} + a^{2} = b * a$.Therefore,$a * b = b * a$ for all $a, b \in Q$.Thus,the binary operation $*$ is commutative.

Class 12 Mathematics Relation and Function Binary Operation

Medium

Let $$ be a binary operation defined on the set of rational numbers $Q$. Determine whether the binary operation defined by $a b = a^{2} + b^{2}$ for all $a, b \in Q$ is commutative.

A
Commutative
B
Not Commutative
C
Associative
D
None of these

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Let $*$ be a binary operation defined on the set of rational numbers $Q$. Determine whether the binary operation defined by $a * b = a^{2} + b^{2}$ for all $a, b \in Q$ is commutative.

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Show that subtraction and division are not binary operations on the set of natural numbers $N$.

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Let $$ be a binary operation defined on the set of rational numbers $Q$. Determine whether the binary operation defined by $a b = a^{2} + b^{2}$ for all $a, b \in Q$ is commutative.

Let $^$ be the binary operation on $N$ defined by $a \,^\, b = \text{H.C.F. of } a \text{ and } b$. Is $^$ commutative? Is $^$ associative? Does there exist an identity for this binary operation on $N$?

On the set $Z$ of all integers,the operation $$ is defined by $a b = a + b - 5$. If $2 * (x * 3) = 5$,then $x$ is equal to:

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