If * is a binary operation defined on R by a | vedclass

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

Question

(A) Given the binary operation $a * b = 1 + ab$ for all $a, b \in R$. $1$. Commutativity: $a * b = 1 + ab$ $b * a = 1 + ba = 1 + ab$ Since $a * b = b

Vedclass · Accepted Answer

$(i)$ Commutative but not associative.

Vedclass · Answer

(A) Given the binary operation $a * b = 1 + ab$ for all $a, b \in R$. $1$. Commutativity: $a * b = 1 + ab$ $b * a = 1 + ba = 1 + ab$ Since $a * b = b * a$,the operation $*$ is commutative. $2$. Associativity: Consider $a * (b * c) = a * (1 + bc) = 1 + a(1 + bc) = 1 + a + abc$. Now consider $(a * b) * c = (1 + ab) * c = 1 + (1 + ab)c = 1 + c + abc$. Since $1 + a + abc 
eq 1 + c + abc$ for all $a, b, c \in R$,the operation $*$ is not associative. Therefore,the operation $*$ is commutative but not associative.

If $$ is a binary operation defined on $R$ by $a b = 1 + ab, \forall a, b \in R$. Then the operation $*$ is:

Similar Questions

Let $A = \{0, 1, 2, 3, 4, 5, 6\}$. If $a, b \in A$,and $a * b$ is defined as the remainder when $ab$ is divided by $7$,then find the inverse of $2$ under the operation $*$.

Is $^$ defined on the set $\{1, 2, 3, 4, 5\}$ by $a \,^\, b = \text{L.C.M. of } a \text{ and } b$ a binary operation? Justify your answer.

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

Show that $: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ defined by $a b = a + 2b$ is not commutative.

Let $^$ be a binary operation on the set $Q$ of rational numbers defined as $a \,^ \,b = a^{2} + b^{2}$. Which of the following is true?

Vedclass Test Series

Exam Paper Generator

Online Exam Module

If $*$ is a binary operation defined on $R$ by $a * b = 1 + ab, \forall a, b \in R$. Then the operation $*$ is: