Class 12 Mathematics Relations and Functions Binary Operation

Medium

Show that $: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ defined by $a b = a + 2b$ is not commutative.

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Solution

(N/A) To check if the operation $*$ is commutative,we need to verify if $a * b = b * a$ for all $a, b \in \mathbb{R}$.
Given $a * b = a + 2b$.
Let us take two real numbers,$a = 3$ and $b = 4$.
Then,$a * b = 3 * 4 = 3 + 2(4) = 3 + 8 = 11$.
Now,calculate $b * a = 4 * 3 = 4 + 2(3) = 4 + 6 = 10$.
Since $a * b \neq b * a$ (because $11 \neq 10$),the operation $*$ is not commutative.

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Binary Operation

Show that $+: R \times R \rightarrow R$ and $\times: R \times R \rightarrow R$ are commutative binary operations,but $-: R \times R \rightarrow R$ and $\div: R_* \times R_* \rightarrow R_*$ are not commutative.

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Show that $0$ is the identity for addition on $R$ and $1$ is the identity for multiplication on $R$. But there is no identity element for the operations $-: R \times R \rightarrow R$ and $\div : R_* \times R_* \rightarrow R_*$.

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

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Show that none of the operations given above has an identity element.

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Let $A = N \times N$ and $^$ be the binary operation on $A$ defined by $(a, b) \,^\, (c, d) = (a + c, b + d)$. Determine whether the operation $^*$ is commutative,associative,and has an identity element.

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Show that $*: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ defined by $a * b = a + 2b$ is not commutative.

Similar Questions

Show that $+: R \times R \rightarrow R$ and $\times: R \times R \rightarrow R$ are commutative binary operations,but $-: R \times R \rightarrow R$ and $\div: R_* \times R_* \rightarrow R_*$ are not commutative.

Show that $0$ is the identity for addition on $R$ and $1$ is the identity for multiplication on $R$. But there is no identity element for the operations $-: R \times R \rightarrow R$ and $\div : R_* \times R_* \rightarrow R_*$.

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

Show that none of the operations given above has an identity element.

Let $A = N \times N$ and $^*$ be the binary operation on $A$ defined by $(a, b) \,^*\, (c, d) = (a + c, b + d)$. Determine whether the operation $^*$ is commutative,associative,and has an identity element.

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Show that $: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ defined by $a b = a + 2b$ is not commutative.

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

Let $A = N \times N$ and $^$ be the binary operation on $A$ defined by $(a, b) \,^\, (c, d) = (a + c, b + d)$. Determine whether the operation $^*$ is commutative,associative,and has an identity element.