Class 12 Mathematics Relation and Function Binary Operation

Medium

Show that addition and multiplication are associative binary operations on $R$. However,subtraction is not associative on $R$,and division is not associative on $R_*$.

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

Vedclass pdf generator app on play store

Solution

For any $a, b, c \in R$,addition is associative because $(a + b) + c = a + (b + c)$.
Multiplication is associative because $(a \times b) \times c = a \times (b \times c)$.
Subtraction is not associative because $(8 - 5) - 3 = 3 - 3 = 0$,while $8 - (5 - 3) = 8 - 2 = 6$. Since $0 \neq 6$,subtraction is not associative.
Division is not associative on $R_*$ (where $R_* = R \setminus \{0\}$) because $(8 \div 5) \div 3 = \frac{8}{5} \div 3 = \frac{8}{15}$,while $8 \div (5 \div 3) = 8 \div \frac{5}{3} = \frac{24}{5}$. Since $\frac{8}{15} \neq \frac{24}{5}$,division is not associative.

Explore More

Browse All

Question Bank

All Subjects

Class 12 Questions

Class 12

Mathematics Questions

Chapter

Relation and Function

Subtopic

Binary Operation

Consider the binary operations $^: R \times R \rightarrow R$ and $o: R \times R \rightarrow R$ defined as $a \,^\, b = |a-b|$ and $a \,o\, b = a$,$\forall \, a, b \in R$. Show that $^$ is commutative but not associative,and $o$ is associative but not commutative. Further,show that $\forall \, a, b, c \in R, a \,^\, (b \,o\, c) = (a \,^\, b) \,o\, (a \,^\, c)$. [If it is so,we say that the operation $^$ distributes over the operation $o$]. Does $o$ distribute over $^$? Justify your answer.

Difficult

View Solution

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 $A^{-1} = A$.

Difficult

View Solution

Let $^$ be the binary operation on $N$ defined by $a \,^\, b = \text{H.C.F. of } a \text{ and } b$. Is $^$ commutative? Is $^$ associative? Does there exist an identity for this binary operation on $N$?

Difficult

View Solution

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

Medium

View Solution

Determine whether or not each of the definitions of $$ given below gives a binary operation. In the event that $$ is not a binary operation,give justification for this. On $Z^{+}$,define $$ by $a b = a$.

Medium

View Solution

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial

For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free

For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo

Show that addition and multiplication are associative binary operations on $R$. However,subtraction is not associative on $R$,and division is not associative on $R_*$.

Similar Questions

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 $A^{-1} = A$.

Let $^*$ be the binary operation on $N$ defined by $a \,^*\, b = \text{H.C.F. of } a \text{ and } b$. Is $^*$ commutative? Is $^*$ associative? Does there exist an identity for this binary operation on $N$?

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

Determine whether or not each of the definitions of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation,give justification for this. On $Z^{+}$,define $*$ by $a * b = a$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module

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 $A^{-1} = A$.

Let $^$ be the binary operation on $N$ defined by $a \,^\, b = \text{H.C.F. of } a \text{ and } b$. Is $^$ commutative? Is $^$ associative? Does there exist an identity for this binary operation on $N$?

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

Determine whether or not each of the definitions of $$ given below gives a binary operation. In the event that $$ is not a binary operation,give justification for this. On $Z^{+}$,define $$ by $a b = a$.