Show that the operation $*: R \times R \rightarrow R$ defined by $a * b = a + 2b$ is not associative.
(N/A) An operation $*$ is associative if $(a * b) * c = a * (b * c)$ for all $a, b, c \in R$.
First,calculate $(a * b) * c$:
$(a * b) * c = (a + 2b) * c = (a + 2b) + 2c = a + 2b + 2c$.
Next,calculate $a * (b * c)$:
$a * (b * c) = a * (b + 2c) = a + 2(b + 2c) = a + 2b + 4c$.
Since $a + 2b + 2c \neq a + 2b + 4c$ for all $a, b, c \in R$,the operation is not associative.
For example,let $a = 8, b = 5, c = 3$:
$(8 * 5) * 3 = (8 + 2(5)) * 3 = 18 * 3 = 18 + 2(3) = 18 + 6 = 24$.
$8 * (5 * 3) = 8 * (5 + 2(3)) = 8 * (5 + 6) = 8 * 11 = 8 + 2(11) = 8 + 22 = 30$.
Since $24 \neq 30$,the operation is not associative.
First,calculate $(a * b) * c$:
$(a * b) * c = (a + 2b) * c = (a + 2b) + 2c = a + 2b + 2c$.
Next,calculate $a * (b * c)$:
$a * (b * c) = a * (b + 2c) = a + 2(b + 2c) = a + 2b + 4c$.
Since $a + 2b + 2c \neq a + 2b + 4c$ for all $a, b, c \in R$,the operation is not associative.
For example,let $a = 8, b = 5, c = 3$:
$(8 * 5) * 3 = (8 + 2(5)) * 3 = 18 * 3 = 18 + 2(3) = 18 + 6 = 24$.
$8 * (5 * 3) = 8 * (5 + 2(3)) = 8 * (5 + 6) = 8 * 11 = 8 + 2(11) = 8 + 22 = 30$.
Since $24 \neq 30$,the operation is not associative.
Explore More
Similar Questions
Let $^*$ be the binary operation on $N$ given by $a \, ^* \, b = \text{L.C.M. of } a \text{ and } b$. Is $^*$ commutative?
Medium
View SolutionOn the set of all non-zero reals,an operation $*$ is defined as $a * b = \frac{3ab}{2}$. In this group,a solution of $(2 * x) * 3^{-1} = 4^{-1}$ is
MediumKCET 2011
View SolutionWhich of the following is a subgroup of the group $G = \{2^{n} \mid n \in \mathbb{Z}\}$ under multiplication?
MediumKCET 2012
View SolutionShow that $-a$ is the inverse of $a$ for the addition operation '$+$' on $R$ and $\frac{1}{a}$ is the inverse of $a \neq 0$ for the multiplication operation '$\times$' on $R$.
Medium
View SolutionConsider 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)
$^*$ $1$ $2$ $3$ $4$ $5$ $1$ $1$ $1$ $1$ $1$ $1$ $2$ $1$ $2$ $2$ $2$ $2$ $3$ $1$ $2$ $3$ $3$ $3$ $4$ $1$ $2$ $3$ $4$ $4$ $5$ $1$ $2$ $3$ $4$ $5$
(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 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