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

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

Question

(NONE) On $Z^+$,the operation $^*$ is defined by $a ^* b = a^b$. $1$. Commutativity: We check if $a ^* b = b ^* a$ for all $a, b \in Z^+$. Consider $a

Vedclass · Accepted Answer

Vedclass · Answer

(NONE) On $Z^+$,the operation $^*$ is defined by $a ^* b = a^b$. $1$. Commutativity: We check if $a ^* b = b ^* a$ for all $a, b \in Z^+$. Consider $a = 1$ and $b = 2$. $1 ^* 2 = 1^2 = 1$ $2 ^* 1 = 2^1 = 2$ Since $1 
eq 2$,$1 ^* 2 
eq 2 ^* 1$. Therefore,the operation $^*$ is not commutative. $2$. Associativity: We check if $(a ^* b) ^* c = a ^* (b ^* c)$ for all $a, b, c \in Z^+$. Consider $a = 2, b = 3, c = 4$. $(2 ^* 3) ^* 4 = (2^3) ^* 4 = 8 ^* 4 = 8^4 = (2^3)^4 = 2^{12}$. $2 ^* (3 ^* 4) = 2 ^* (3^4) = 2 ^* 81 = 2^{81}$. Since $2^{12} 
eq 2^{81}$,$(2 ^* 3) ^* 4 
eq 2 ^* (3 ^* 4)$. Therefore,the operation $^*$ is not associative.

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

Similar Questions

Let $^$ be the binary operation on $N$ given by $a \,^ \,b = \text{L.C.M. of } a \text{ and } b$. Find $5 \,^* \,7$ and $20 \,^* \,16$.

In a group $(G, *)$,for some element $a$ of $G$,if $a^{2}=e$,where $e$ is the identity element,then

For any two real numbers,an operation $$ defined by $a b = 1 + ab$ is

On the set of positive rationals,a binary operation $$ is defined by $a b = \frac{2ab}{5}$. If $2 * x = 3^{-1}$,then $x = $

Let $^$ be a binary operation on the set $Q$ of rational numbers defined as $a ^ b = a - b$. Determine whether the operation $^*$ is commutative and associative.

Vedclass Test Series

Exam Paper Generator

Online Exam Module

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