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.

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

Similar Questions

Show that the $\vee: R \times R \rightarrow R$ given by $(a, b) \rightarrow \max\{a, b\}$ and the $\wedge: R \times R \rightarrow R$ given by $(a, b) \rightarrow \min\{a, b\}$ are binary operations.

On the set $Z$ of all integers,the operation $$ is defined by $a b = a + b - 5$. If $2 * (x * 3) = 5$,then $x$ is equal to:

The inverse of $2010$ in the group $Q^{+}$ of all positive rational numbers under the binary operation $$ defined by $a b = \frac{ab}{2010}, \forall a, b \in Q^{+}$,is

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.

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

Vedclass Test Series

Exam Paper Generator

Online Exam Module

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