CBSE Mathematics for Class XII

Dinesh Khattar; Anita Khattar

Chapter 4

Relations and Functions - all with Video Answers

Educators

Section 1

Review Of Concepts

Problem 1

Check whether the relation $R$ defined in the set $\{1,2,3,4,5,6\}$ as $R=\{(a, b): b=a+1\}$ is reflexive, symmetric or transitive.

Anurag Kumar

Anurag Kumar

Numerade Educator

Problem 2

Let $L$ be the set of all lines in a plane and $R$ be the relation in $L$ defined as $R=\left\{\left(L_{1}, L_{2}\right): L_{1}\right.$ is perpendicular to $\left.L_{2}\right\}$. Show that $R$ is symmetric, but neither reflexive nor transitive.

Anurag Kumar

Numerade Educator

Problem 3

Determine whether each of the following relations are reflexive, symmetric and transitive:
(i) Relation $R$ in the set $A=\{1,2,3, \ldots, 13,14\}$ defined as:
$$
R=\{(x, y): 3 x-y=0\}
$$
(ii) Relation $R$ in the set $N$ of natural numbers defined as:
$$
R=\{(x, y): y=x+5 \text { and } x<4\}
$$
(iii) Relation $R$ in the set $A=\{1,2,3,4,5,6\}$ as:
$R=\{(x, y): y$ is divisible by $x\}$
(iv) Relation $R$ in the set $Z$ of all integers defined as:
$R=\{(x, y): x-y$ is an integer $\}$
(v) Relation $R$ in the set $A$ of human beings in a town at a particular time given by
(a) $R=\{(x, y): x$ and $y$ work at the same place $\}$
(b) $R=\{(x, y): x$ and $y$ live in the same locality $\}$
(c) $R=\{(x, y): x$ is exactly $7 \mathrm{~cm}$ taller than $y\}$
(d) $R=\{(x, y): x$ is wife of $y\}$
(e) $R=\{(x, y): x$ is father of $y\}$

Anurag Kumar

Numerade Educator

Problem 4

Show that the relation $R$ in the set $\{1,2,3\}$ given by $R=\{(1,1),(2,2),(3,3),(1,2),(2,3)\}$ is reflexive, but neither symmetric nor transitive.

Anurag Kumar

Numerade Educator

Problem 5

Show that the relation $R$ in $\mathbf{R}$ defined as $R=\{(a, b): a \leq b\}$ is reflexive and transitive, but not symmetric.

Anurag Kumar

Numerade Educator

Problem 6

Show that the relation $R$ in the set $Z$ of integers given by
$$
R=\{(a, b): 2 \text { divides } a-b\}
$$
is an equivalence relation.

Clayton Schubring

Clayton Schubring

Numerade Educator

Problem 7

Show that the relation $R$ in the set $\mathbf{R}$ of real numbers, defined as $R=\left\{(a, b): a \leq b^{2}\right\}$ is neither reflexive nor symmetric nor transitive.

Anurag Kumar

Numerade Educator

Problem 8

Let $R$ be the relation defined in the set $A=\{1,2,3,4,5,6,7\}$ by $R=\{(a, b):$ both $a$ and $b$ are either odd or even $\}$. Show that $R$ is an equivalence relation. Further, show that all the elements of the subset $\{1,3,5,7\}$ are related to each other and all the elements of the subset $\{2,4,6\}$ are related to each other, but no element of the subset $\{1,3,5,7\}$ is related to any element of the subset $\{2,4,6\}$

Anurag Kumar

Numerade Educator

Problem 9

Give an example of a relation, which is
(i) Symmetric, but neither reflexive nor transitive.
(ii) Transitive, but neither reflexive nor symmetric.
(iii) Reflexive and symmetric, but not transitive.
(iv) Reflexive and transitive, but not symmetric.
(v) Symmetric and transitive, but not reflexive.

Anurag Kumar

Numerade Educator

Problem 10

Show that the relation $R$ in the set $A$ of all the books in a library of a college, given by $R=\{(x, y)$ $: x$ and $y$ have same number of pages $\}$ is an equivalence relation.

Anurag Kumar

Numerade Educator

Problem 11

Show that the relation $R$ defined in the set $A$ of all triangles as $R=\left\{\left(T_{1}, T_{2}\right): T_{1}\right.$ is similar to $\left.T_{2}\right\}$, is equivalence relation. Consider three right angle, triangles $T_{1}$ with sides $3,4,5, T_{2}$ with sides 5 , 12,13 and $T_{3}$ with sides $6,8,10$. Which triangles among $T_{1}, T_{2}$ and $T_{3}$ are related?

Anurag Kumar

Numerade Educator

Problem 12

Let $R$ be the relation in the set $\{1,2,3,4\}$ given by $R=\{(1,2),(2,2),(1,1),(4,4),(1,3),(3,3)$, $(3,2)\}$. Choose the correct answer.
(a) $R$ is reflexive and symmetric, but not transitive.
(b) $R$ is reflexive and transitive, but not symmetric.
(c) $R$ is symmetric and transitive, but not reflexive.
(d) $R$ is an equivalence relation.

Anurag Kumar

Numerade Educator

Problem 13

Show that the relation $R$ in the set $A$ of points in a plane given by $R=\{(P, Q):$ distance of the point $P$ from the origin is same as the distance of the point $Q$ from the origin $\},$ is an equivalence relation. Further, show that the set of all points related to a point $P \neq(0,0)$ is the circle passing through $P$ with origin as centre.

Anurag Kumar

Numerade Educator

Problem 14

Check whether the relation $R$ in $\mathbf{R}$ defined by $R=\left\{(a, b): a \leq b^{3}\right\}$ is reflexive, symmetric or transitive.

Anurag Kumar

Numerade Educator

Problem 15

Show that the relation $R$ in the set $A=\{1,2,3,4,5\}$ given by $R=\{(a, b):|a-b|$ is even $\}$, is an equivalence relation. Show that all the elements of $\{1,3,5\}$ are related to each other and all the elements of $\{2,4\}$ are related to each other. But no element of $\{1,3,5\}$ is related to any element of $\{2,4\}$.

Anurag Kumar

Numerade Educator

Problem 16

Let $R$ be the relation in the set $N$ given by $R=\{(a, b): a=b-2, b>6\}$. Choose the correct answer.
(a) $(2,4) \in R$
(b) $(3,8) \in R$
(c) $(6,8) \in R$
(d) $(8,7) \in R$

Anurag Kumar

Numerade Educator

Problem 17

Let $L$ be the set of all lines in $X Y$ plane and $R$ be the relation in $L$ defined as $R=\left\{\left(L_{1}, L_{2}\right): L_{1}\right.$ is parallel to $\left.L_{2}\right\}$. Show that $R$ is an equivalence relation. Find the set of all lines related to the line $y=2 x+4$.

Anurag Kumar

Numerade Educator

Problem 18

Show that each of the relation $R$ in the set $A=\{x \in Z: 0 \leq x \leq 12\}$, given by
(i) $R=\{(a, b):|a-b|$ is a multiple of 4$\}$
(ii) $R=\{(a, b): a=b\}$
is an equivalence relation. Find the set of all elements related to 1 in each case.

Anurag Kumar

Numerade Educator

Problem 19

Show that the relation $R$ in the set $\{1,2,3\}$ given $R=\{(1,2),(2,1)\}$ is symmetric, but neither reflexive nor transitive.

Anurag Kumar

Numerade Educator

Problem 20

Show that the relation $R$ defined in the set $A$ of all polygons as $R=\left\{\left(P_{1}, P_{2}\right),: P_{1}\right.$ and $P_{2}$ have same number of sides $\}$, is an equivalence relation. What is the set of all elements in $A$ related to the right angled triangle $T$ with sides 3,4 and 5 ?

Anurag Kumar

Numerade Educator

Problem 21

Let $R$ be a relation on the set $A$ of ordered pairs of positive integers defined by $(x, y) R(u, v)$ if and only if $x v=y u$. Show that $R$ is an equivalence relation.

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 22

Given a non empty set $X$, consider $P(X)$ which is the set of all subsets of $X$. Define the relation $R$ in $P(X)$ as follows:
For subsets $A, B$ in $P(X), A R B$ if and only if $A \subset B$. Is $R$ an equivalence relation on $P(X) ?$ Justify your answer.

Anurag Kumar

Numerade Educator

Problem 23

Let $X=\{1,2,3,4,5,6,7,8,9\} .$ Let $R_{1}$ be a relation in $X$ given by $R_{1}=\{(x, y): x-y$ is divisible by 3$\}$ and $R_{2}$ be another relation on $X$ given by $R_{2}=\{(x, y):(x, y) \subset\{1,4,7\}\}$ or $\{x, y\} \subset\{2,5,8\}$ or $\{x, y\} \subset\{3,6,9\}\} .$ Show that $R_{1}=R_{2}$.

Anurag Kumar

Numerade Educator

Problem 24

If $R_{1}$ and $R_{2}$ are equivalence relations in a set $A$, show that $R_{1} \cap R_{2}$ is also an equivalence relation.

Muhammad Nawaz

Numerade Educator

Problem 25

Let $A=\{1,2,3\}$. Then number of equivalence relations containing $(1,2)$ is:
(a) 1
(b) 2
(c) 3
(d) 4

Gaurav Kalra

Numerade Educator

Problem 26

Let $A=\{1,2,3\}$. Then show that the number of relations containing $(1,2)$ and $(2,3)$ which are reflexive and transitive, but not symmetric is four.

Anurag Kumar

Numerade Educator

Problem 27

Show that the number of equivalence relations in the set $\{1,2,3\}$ containing $(1,2)$ and $(2,1)$ is two.

Ibrahima Barry

Numerade Educator

Problem 28

Show that the number of equivalence relations in the set $\{1,2,3\}$ containing $(1,2)$ and $(2,1)$ is two.

Ibrahima Barry

Numerade Educator

Problem 29

Following are some relations on set $A=\{$ Students of some boys school $\} .$ Write, which type of following relations are:
$R_{1}=\{(a, b): a, b$ are ages of students and $|a-b| \geq 0\}$ $R_{2}=\{(a, b): a, b$ are heights of students and $|a-b|<0\}$ $R_{3}=\{(a, b): a, b$ are weights of students and $|a-b|>0\}$ $R_{4}=\{(a, b): a$ and $b$ are studying in the same class $\}$ $R_{5}=\{(a, b): a, b$ are studying in the same school $\}$.

James Chok

Numerade Educator

Problem 30

Check the following relations for each of:
(i) Reflexivity
(ii) Symmetricity
(iii) Transitivity
(iv) Equivalence
(a) $R_{1}=\left\{\left(l_{1}, l_{2}\right):\left|l_{1}\right|=\left|l_{2}\right|, l_{1}\right.$ and $l_{2}$ are line segments in the same plane\}
(b) $R_{2}=\{(a, b),(b, b),(c, c),(a, c),(b, c)\}$ in the set $\{a, b, c\}$
(c) $R_{3}=\{(a, b): a \geq b, a \in R, b \in R\}$
(d) $R_{4}=\{(a, b): a$ is a divisor of $b, a \in A, b \in A$ where $A=\{1,2,4\}\}$
(e) $R_{5}=\{(a, b): a=b+1, a \in R, b \in R\}$
(f) $R_{6}=\{(a, b)(b, a)\}$ in the set $\{a, b, c\}$
(g) $R_{7}=\{(a, b): a \geq b, a \in N, b \in N\}$
(h) $R_{8}=\{(a, b): a>b, a \in N, b \in N\}$
(i) $R_{9}=\{(a, b): a$ is a divisor of $b, a \in N, b \in N\}$
(j) $R_{10}=\{(a, b): a$ is divisible by $b, a \in N, b \in N\}$
(k) $R_{11}=\{(a, b): a \leq b, a \in Z, b \in Z\}, Z$ is the set of integers
(1) $R_{12}=\left\{(a, b): a=b^{3}, a \in Z, b \in Z\right\}, Z$ is the set of integers
$(\mathrm{m}) R_{13}=\{(a, b): b=3 a, a \in R, b \in R\}$
(n) $R_{14}=\{(a, b): a-b$ is a multiple of $5, a \in R, b \in R\}$
(o) $R_{15}=\{(a, b): a-b$ is an integer, $a \in R, b \in R\}$

Anurag Kumar

Numerade Educator