Consider a binary operation * on the set {1,2 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) binary operation $*$ on a set $S$ is commutative if $a \,^*\, b = b \,^*\, a$ for all $a, b \in S$.To check if the operation is commutative,we che

Vedclass · Accepted Answer

Vedclass · Answer

(A) binary operation $*$ on a set $S$ is commutative if $a \,^*\, b = b \,^*\, a$ for all $a, b \in S$.
To check if the operation is commutative,we check if the table is symmetric with respect to the main diagonal.
Looking at the table:
- $1 \,^*\, 2 = 1$ and $2 \,^*\, 1 = 1$. Thus,$1 \,^*\, 2 = 2 \,^*\, 1$.
- $1 \,^*\, 3 = 1$ and $3 \,^*\, 1 = 1$. Thus,$1 \,^*\, 3 = 3 \,^*\, 1$.
- $2 \,^*\, 3 = 2$ and $3 \,^*\, 2 = 2$. Thus,$2 \,^*\, 3 = 3 \,^*\, 2$.
- $2 \,^*\, 4 = 2$ and $4 \,^*\, 2 = 2$. Thus,$2 \,^*\, 4 = 4 \,^*\, 2$.
- $3 \,^*\, 4 = 3$ and $4 \,^*\, 3 = 3$. Thus,$3 \,^*\, 4 = 4 \,^*\, 3$.
- $1 \,^*\, 5 = 1$ and $5 \,^*\, 1 = 1$. Thus,$1 \,^*\, 5 = 5 \,^*\, 1$.
- $2 \,^*\, 5 = 2$ and $5 \,^*\, 2 = 2$. Thus,$2 \,^*\, 5 = 5 \,^*\, 2$.
- $3 \,^*\, 5 = 3$ and $5 \,^*\, 3 = 3$. Thus,$3 \,^*\, 5 = 5 \,^*\, 3$.
- $4 \,^*\, 5 = 4$ and $5 \,^*\, 4 = 4$. Thus,$4 \,^*\, 5 = 5 \,^*\, 4$.
Since $a \,^*\, b = b \,^*\, a$ for all $a, b \in \{1,2,3,4,5\}$,the operation $^*$ is commutative.

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

Similar Questions

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 $a b = a$.

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 $a b = |a - b|$.

For each binary operation $^$ defined below,determine whether $^$ is commutative or associative. On $Z$,define $a ^* b = a - b$.

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

Let $^$ be a binary operation on the set $Q$ of rational numbers defined as $a b = (a - b)^2$. Determine whether the operation is commutative and associative.

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Similar Questions

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 $a * b = a$.

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 $a * b = |a - b|$.

For each binary operation $^*$ defined below,determine whether $^*$ is commutative or associative. On $Z$,define $a ^* b = a - b$.