(A) binary operation $*$ is commutative if $a * b = b * a$ for all $a, b \in Q$. Given $a * b = (a - b)^{2}$. Now,calculate $b * a$: $b * a = (b - a)^

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(A) binary operation $*$ is commutative if $a * b = b * a$ for all $a, b \in Q$. Given $a * b = (a - b)^{2}$. Now,calculate $b * a$: $b * a = (b - a)^{2}$. Since $(b - a)^{2} = [-(a - b)]^{2} = (-1)^{2}(a - b)^{2} = (a - b)^{2}$. Therefore,$b * a = (a - b)^{2} = a * b$. Since $a * b = b * a$ for all $a, b \in Q$,the 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 $a b = (a - b)^{2}$ for all $a, b \in Q$ is commutative.

A
Commutative
B
Not commutative
C
Associative
D
None of these

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Class 12 Questions

Class 12

Mathematics Questions

Chapter

Relation and Function

Subtopic

Binary Operation

Consider a binary operation $$ on the set $\{1, 2, 3, 4, 5\}$ given by the following multiplication table. Compute $(2 \,^ \,3) \,^* \,(4 \,^* \,5)$.
(Hint: use the following table)

$^*$ $1$ $2$ $3$ $4$ $5$

$1$ $1$ $1$ $1$ $1$ $1$

$2$ $1$ $2$ $2$ $2$ $2$

$3$ $1$ $2$ $3$ $3$ $3$

$4$ $1$ $2$ $3$ $4$ $4$

$5$ $1$ $2$ $3$ $4$ $5$

Medium

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Let $^$ be a binary operation on the set $Q$ of rational numbers defined as $a ^ b = a - b$. Determine whether the operation $^*$ is commutative and associative.

Medium

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Let $^$ be the binary operation on $N$ given by $a ^ b = \text{L.C.M. of } a \text{ and } b$. Find which elements of $N$ are invertible for the operation $^*$?

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Determine whether or not the definition of $$ given below gives a binary operation. In the event that $$ is not a binary operation,give justification for this. On $Z^{+}$,define $$ by $a b = a - b$.

Medium

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Which of the following is not a group with respect to the given operation?

EasyKCET 2007

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$^*$	$1$	$2$	$3$	$4$	$5$
$1$	$1$	$1$	$1$	$1$	$1$
$2$	$1$	$2$	$2$	$2$	$2$
$3$	$1$	$2$	$3$	$3$	$3$
$4$	$1$	$2$	$3$	$4$	$4$
$5$	$1$	$2$	$3$	$4$	$5$

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

Similar Questions

Let $^*$ be a binary operation on the set $Q$ of rational numbers defined as $a ^* b = a - b$. Determine whether the operation $^*$ is commutative and associative.

Let $^*$ be the binary operation on $N$ given by $a ^* b = \text{L.C.M. of } a \text{ and } b$. Find which elements of $N$ are invertible for the operation $^*$?

Determine whether or not the definition of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation,give justification for this. On $Z^{+}$,define $*$ by $a * b = a - b$.

Which of the following is not a group with respect to the given operation?

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

Let $^$ be a binary operation on the set $Q$ of rational numbers defined as $a ^ b = a - b$. Determine whether the operation $^*$ is commutative and associative.

Let $^$ be the binary operation on $N$ given by $a ^ b = \text{L.C.M. of } a \text{ and } b$. Find which elements of $N$ are invertible for the operation $^*$?

Determine whether or not the definition of $$ given below gives a binary operation. In the event that $$ is not a binary operation,give justification for this. On $Z^{+}$,define $$ by $a b = a - b$.