State whether the following statement is true or false and justify your answer: If $^*$ is a commutative binary operation on $N$,then $a ^* (b ^* c) = (c ^* b) ^* a$.
(A) Given that $^*$ is a commutative binary operation on $N$,which means $x ^* y = y ^* x$ for all $x, y \in N$.
Consider the right-hand side $(RHS)$:
$RHS = (c ^* b) ^* a$
Since $^*$ is commutative,we can write $(c ^* b) = (b ^* c)$.
So,$RHS = (b ^* c) ^* a$.
Now,by the commutative property,we can swap the elements around the operation $^*$:
$(b ^* c) ^* a = a ^* (b ^* c)$.
Thus,$RHS = a ^* (b ^* c) = LHS$.
Therefore,the statement is true.
Consider the right-hand side $(RHS)$:
$RHS = (c ^* b) ^* a$
Since $^*$ is commutative,we can write $(c ^* b) = (b ^* c)$.
So,$RHS = (b ^* c) ^* a$.
Now,by the commutative property,we can swap the elements around the operation $^*$:
$(b ^* c) ^* a = a ^* (b ^* c)$.
Thus,$RHS = a ^* (b ^* c) = LHS$.
Therefore,the statement is true.
Explore More
Similar Questions
If $a * b = 10ab$ on $Q^{+}$,then find the inverse of $0.01$.
Medium
View SolutionConsider a binary operation $*$ on the set $\{1,2,3,4,5\}$ given by the following multiplication table. Is $^*$ commutative?
(Hint: use the following table)
$^*$ $1$ $2$ $3$ $4$ $5$ $1$ $1$ $1$ $1$ $1$ $1$ $2$ $1$ $2$ $2$ $2$ $2$ $3$ $1$ $2$ $3$ $3$ $3$ $4$ $1$ $2$ $3$ $4$ $4$ $5$ $1$ $2$ $3$ $4$ $5$
(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
View SolutionLet $*$ be a binary operation defined on $R$ by $a * b = \frac{a+b}{4}$ for all $a, b \in R$. Then the operation $*$ is:
DifficultKCET 2016
View SolutionLet $^*$ be the binary operation on $N$ given by $a \, ^* \, b = \text{L.C.M. of } a \text{ and } b$. Find the identity of $^*$ in $N$.
Medium
View SolutionLet $*$ be a binary operation on the set $Q$ of rational numbers defined as $a * b = \frac{ab}{4}$. Which of the following is true?
Medium
View SolutionVedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo