For each binary operation ^* defined below,de | vedclass

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

Question

(A) On $Q$,the operation $^*$ is defined by $a ^* b = ab + 1$.For commutativity:We know that $ab = ba$ for all $a, b \in Q$.Therefore,$ab + 1 = ba + 1

Vedclass · Accepted Answer

$^*$ is commutative but not associative.

Vedclass · Answer

(A) On $Q$,the operation $^*$ is defined by $a ^* b = ab + 1$.For commutativity:We know that $ab = ba$ for all $a, b \in Q$.Therefore,$ab + 1 = ba + 1$ for all $a, b \in Q$.This implies $a ^* b = b ^* a$ for all $a, b \in Q$.Thus,the operation $^*$ is commutative.For associativity:We check if $(a ^* b) ^* c = a ^* (b ^* c)$ for all $a, b, c \in Q$.Consider $(1 ^* 2) ^* 3 = (1 	imes 2 + 1) ^* 3 = 3 ^* 3 = 3 	imes 3 + 1 = 10$.Consider $1 ^* (2 ^* 3) = 1 ^* (2 	imes 3 + 1) = 1 ^* 7 = 1 	imes 7 + 1 = 8$.Since $(1 ^* 2) ^* 3 
eq 1 ^* (2 ^* 3)$,the operation $^*$ is not associative.

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

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

For each binary operation $^$ defined below,determine whether $^$ is commutative or associative. On $Q$,define $a ^* b = ab + 1$.

Similar Questions

Let $^$ be the binary operation on $N$ given by $a \,^\, b = \text{L.C.M. of } a \text{ and } b$. Is $^*$ associative?

For each binary operation $^$ defined below,determine whether $^$ is commutative or associative. On $Q$,define $a ^* b = \frac{ab}{2}$.

On the set of all natural numbers $N$,which one of the following $*$ is a binary operation?

Vedclass Test Series

Exam Paper Generator

Online Exam Module

For each binary operation $^*$ defined below,determine whether $^*$ is commutative or associative. On $Q$,define $a ^* b = ab + 1$.