Show that the operation *: R times R rightarr | vedclass

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

Question

(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

Vedclass · Accepted Answer

Vedclass · Answer

(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 $a + 2b + 2c 
eq a + 2b + 4c$ 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 $24 
eq 30$,the operation is not associative.

Show that the operation $: R \times R \rightarrow R$ defined by $a b = a + 2b$ is not associative.

Similar Questions

State whether the following statement is true or false and justify your answer: If $^$ is a commutative binary operation on $N$,then $a ^ (b ^* c) = (c ^* b) ^* a$.

Which of the following is false?

Given a non-empty set $X$,let $^: P(X) \times P(X) \rightarrow P(X)$ be defined as $A \,^\, B = (A - B) \cup (B - A)$,$\forall A, B \in P(X)$. Show that the empty set $\Phi$ is the identity for the operation $^*$ and all the elements $A$ of $P(X)$ are invertible with $A^{-1} = A$.

Which of the following is a subgroup of the group $G = \{2^{n} \mid n \in \mathbb{Z}\}$ under multiplication?

In the set of integers $(Z, )$,if $a b = a + b - n, \forall a, b \in Z$,where $n$ is a fixed integer,then the inverse of $(-n)$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Show that the operation $*: R \times R \rightarrow R$ defined by $a * b = a + 2b$ is not associative.