Relations and Functions Class 12

Determine whether each of the following relations are reflexive, symmetric and transitive:
(i) Relation R in the set $\mathrm{A}=\{1,2,3, \ldots, 13,14\}$ defined as
$$
\mathrm{R}=\{(x, y): 3 x-y=0\}
$$

Determine whether each of the following relations are reflexive, symmetric and transitive:
(ii) Relation R in the set $\mathbf{N}$ of natural numbers defined as
$$
\mathrm{R}=\{(x, y): y=x+5 \text { and } x<4\}
$$

Determine whether each of the following relations are reflexive, symmetric and transitive:
(iii) Relation R in the set $\mathrm{A}=\{1,2,3,4,5,6\}$ as
$$
\mathrm{R}=\{(x, y): y \text { is divisible by } x\}
$$

Determine whether each of the following relations are reflexive, symmetric and transitive:
(iv) Relation R in the set $\mathbf{Z}$ of all integers defined as
$$
\mathrm{R}=\{(x, y): x-y \text { is an integer }\}
$$

Determine whether each of the following relations are reflexive, symmetric and transitive:
(v) Relation R in the set A of human beings in a town at a particular time given by
(a) $\mathrm{R}=\{(x, y): x$ and $y$ work at the same place $\}$
(b) $\mathrm{R}=\{(x, y): x$ and $y$ live in the same locality $\}$
(c) $\mathrm{R}=\{(x, y): x$ is exactly 7 cm taller than $y\}$
(d) $\mathrm{R}=\{(x, y): x$ is wife of $y\}$
(e) $\mathrm{R}=\{(x, y): x$ is father of $y\}$

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

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

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

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

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

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

Show that the relation R in the set $\mathrm{A}=\{1,2,3,4,5\}$ given by $\mathrm{R}=\{(a, b):|a-b|$ 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\}.

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

Show that each of the relation R in the set A = {x ∈ Z: 0 ≤ x ≤ 12} , given by
(ii) R = {(a, b): a = b} is an equivalence relation. Find the set of all elements related to 1 in each case.

Give an example of a relation. Which is
(i) Symmetric but neither reflexive nor transitive.

Given an example of a relation. Which is
(ii) Transitive but neither reflexive nor symmetric.

Given an example of a relation. Which is
(iii) Reflexive and symmetric but not transitive.

Given an example of a relation. Which is
(iv) Reflexive and transitive but not symmetric.

Given an example of a relation. Which is
(v) Symmetric and transitive but not reflexive.

Show that the relation R in the set A of points in a plane given by $\mathrm{R}=\{(\mathrm{P}, \mathrm{Q})$ : 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 $\mathrm{P} \neq(0,0)$ is the circle passing through P with origin as centre.

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 $\left.\mathrm{T}_2\right\}$, is equivalence relation. Consider three right angle triangles $\mathrm{T}_1$ with sides $3,4,5, \mathrm{~T}_2$ with sides $5,12,13$ and $\mathrm{T}_3$ with sides $6,8,10$. Which triangles among $\mathrm{T}_1, \mathrm{~T}_2$ and $\mathrm{T}_3$ are related?

Show that the relation $R$ defined in the set $A$ of all polygons as $R=\left\{\left(P_1, P_2\right)\right.$ : $P_1$ 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 angle triangle T with sides 3,4 and 5 ?

Let L be the set of all lines in XY plane and R be the relation in L defined as $\mathrm{R}=\left\{\left(\mathrm{L}_1, \mathrm{~L}_2\right): \mathrm{L}_1\right.$ is parallel to $\left.\mathrm{L}_2\right\}$. Show that R is an equivalence relation. Find the set of all lines related to the line $y=2 x+4$.

Let R be the relation in the set $\{1,2,3,4\}$ given by $\mathrm{R}=\{(1,2),(2,2),(1,1),(4,4)$, (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.

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

Show that the function $f: \mathbf{R}_{\bullet} \rightarrow \mathbf{R}_{\bullet}$ defined by $f(x)=\frac{1}{x}$ is one-one and onto, where $\mathbf{R}_{\boldsymbol{*}}$ is the set of all non-zero real numbers. Is the result true, if the domain $\mathbf{R}_{\bullet}$ is replaced by $\mathbf{N}$ with co-domain being same as $\mathbf{R}_{\bullet}$ ?

Check the injectivity and surjectivity of the following functions:
(i) $f: \mathbf{N} \rightarrow \mathbf{N}$ given by $f(x)=x^2$

Check the injectivity and surjectivity of the following functions:
(ii) $f: \mathbf{Z} \rightarrow \mathbf{Z}$ given by $f(x)=x^2$

Check the injectivity and surjectivity of the following functions:
(iii) $f: \mathbf{R} \rightarrow \mathbf{R}$ given by $f(x)=x^2$

Check the injectivity and surjectivity of the following functions:
(iv) $f: \mathbf{N} \rightarrow \mathbf{N}$ given by $f(x)=x^3$

Check the injectivity and surjectivity of the following functions:
(v) $f: \mathbf{Z} \rightarrow \mathbf{Z}$ given by $f(x)=x^3$

Prove that the Greatest Integer Function $f: \mathbf{R} \rightarrow \mathbf{R}$, given by $f(x)=[x]$, is neither one-one nor onto, where $[x]$ denotes the greatest integer less than or equal to $x$.

Show that the Modulus Function $f: \mathbf{R} \rightarrow \mathbf{R}$, given by $f(x)=|x|$, is neither oneone nor onto, where $|x|$ is $x$, if $x$ is positive or 0 and $|x|$ is $-x$, if $x$ is negative.

Show that the Signum Function $f: \mathbf{R} \rightarrow \mathbf{R}$, given by
$$
f(x)=\left\{\begin{array}{r}
1, \text { if } x>0 \\
0, \text { if } x=0 \\
1, \text { if } x<0
\end{array}\right.
$$

is neither one-one nor onto.

Let $\mathrm{A}=\{1,2,3\}, \mathrm{B}=\{4,5,6,7\}$ and let $f=\{(1,4),(2,5),(3,6)\}$ be a function from A to B . Show that $f$ is one-one.

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) $f: \mathbf{R} \rightarrow \mathbf{R}$ defined by $f(x)=3-4 x$

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(ii) $f: \mathbf{R} \rightarrow \mathbf{R}$ defined by $f(x)=1+x^2$

Let A and B be sets. Show that $f: \mathrm{A} \times \mathrm{B} \rightarrow \mathrm{B} \times \mathrm{A}$ such that $f(a, b)=(b, a)$ is bijective function.

Let $f: \mathbf{N} \rightarrow \mathbf{N}$ be defined by $f(n)=\left\{\begin{array}{l}\frac{n+1}{2}, \text { if } n \text { is odd } \\ \frac{n}{2}, \text { if } n \text { is even }\end{array}\right.$ for all $n \in \mathbf{N}$.
State whether the function $f$ is bijective. Justify your answer.

Let $\mathrm{A}=\mathbf{R}-\{3\}$ and $\mathrm{B}=\mathbf{R}-\{1\}$. Consider the function $f: \mathrm{A} \rightarrow \mathrm{B}$ defined by $f(x)=\left(\frac{x-2}{x-3}\right)$. Is $f$ one-one and onto? Justify your answer.

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined as $f(x)=x^4$. Choose the correct answer.
(A) $f$ is one-one onto
(B) $f$ is many-one onto
(C) $f$ is one-one but not onto
(D) $f$ is neither one-one nor onto.

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined as $f(x)=3 x$. Choose the correct answer.
(A) $f$ is one-one onto
(B) $f$ is many-one onto
(C) $f$ is one-one but not onto
(D) $f$ is neither one-one nor onto.

Let $A$ be the set of all students of a boys’ school. Show that the relation $R$ in $A$ given by

$$
R=\{(a,b):a\text{ is the sister of }b\}
$$

is the empty relation and

$$
R'=\{(a,b):\text{the difference between the heights of }a\text{ and }b\text{ is less than }3\text{ metres}\}
$$

is the universal relation.

Let $T$ be the set of all triangles in a plane, with $R$ a relation in $T$ given by

$$
R=\{(T_1,T_2):T_1\text{ is congruent to }T_2\}.
$$

Show that $R$ is an equivalence relation.

Let $L$ be the set of all lines in a plane and $R$ be the relation in $L$ defined as

$$
R=\{(L_1,L_2):L_1\text{ is perpendicular to }L_2\}.
$$

Show that $R$ is symmetric but neither reflexive nor transitive.

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.

Show that the relation $R$ in the set $\mathbb{Z}$ of integers given by

$$
R=\{(a,b):2\text{ divides }a-b\}
$$

is an equivalence relation.

Let $R$ be the relation defined in the set

$$
A=\{1,2,3,4,5,6,7\}
$$

by

$$
R=\{(a,b):\text{both }a\text{ and }b\text{ are either odd or even}\}.
$$

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 $\{2,4,6\}$.

Let $A$ be the set of all $50$ students of Class X in a school. Let

$$
f:A\rightarrow\mathbb{N}
$$

be a function defined by

$$
f(x)=\text{roll number of the student }x.
$$

Show that $f$ is one-one but not onto.

Show that the function

$$
f:\mathbb{N}\rightarrow\mathbb{N}
$$

given by

$$
f(x)=2x
$$

is one-one but not onto.

Prove that the function

$$
f:\mathbb{R}\rightarrow\mathbb{R}
$$

given by

$$
f(x)=2x
$$
is one-one and onto.

Show that the function

$$
f:\mathbb{N}\rightarrow\mathbb{N}
$$

given by

$$
f(1)=f(2)=1
$$

and

$$
f(x)=x-1,\qquad x>2,
$$

is onto but not one-one.

Show that the function

$$
f:\mathbb{R}\rightarrow\mathbb{R}
$$

defined as

$$
f(x)=x^2
$$

is neither one-one nor onto.

Show that

$$
f:\mathbb{N}\rightarrow\mathbb{N},
$$

given by

$$
f(x)=
\begin{cases}
x+1, & \text{if }x\text{ is odd},\\
x-1, & \text{if }x\text{ is even},
\end{cases}
$$

is both one-one and onto.

Show that an onto function

$$
f:\{1,2,3\}\rightarrow\{1,2,3\}
$$

is always one-one.

Show that a one-one function

$$
f:\{1,2,3\}\rightarrow\{1,2,3\}
$$

must be onto.

Let

$$
f:\{2,3,4,5\}\rightarrow\{3,4,5,9\}
$$

and

$$
g:\{3,4,5,9\}\rightarrow\{7,11,15\}
$$

be functions defined as

$$
f(2)=3,\qquad f(3)=4,\qquad f(4)=f(5)=5
$$

and

$$
g(3)=g(4)=7,\qquad g(5)=g(9)=11.
$$

Find

$$
g\circ f.
$$

Find $g\circ f$ and $f\circ g$, if

$$
f:\mathbb{R}\rightarrow\mathbb{R}
$$

and

$$
g:\mathbb{R}\rightarrow\mathbb{R}
$$

are given by

$$
f(x)=\cos x
$$

and

$$
g(x)=3x^2.
$$

Show that

$$
g\circ f\neq f\circ g.
$$

Let

$$
f:\mathbb{N}\rightarrow Y
$$

be a function defined as

$$
f(x)=4x+3,
$$

where

$$
Y=\{y\in\mathbb{N}:y=4x+3\text{ for some }x\in\mathbb{N}\}.
$$

Show that $f$ is invertible. Find the inverse.

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.

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

$$
xv=yu.
$$

Show that $R$ is an equivalence relation.

Let

$$
X=\{1,2,3,4,5,6,7,8,9\}.
$$

Let $R_1$ be a relation in $X$ given by

$$
R_1=\{(x,y):x-y\text{ is divisible by }3\}
$$

and $R_2$ be another relation on $X$ given by

$$
R_2=
\left\{
(x,y):
\{x,y\}\subset\{1,4,7\}
\text{ or }
\{x,y\}\subset\{2,5,8\}
\text{ or }
\{x,y\}\subset\{3,6,9\}
\right\}.
$$

Show that

$$
R_1=R_2.
$$

Let

$$
f:X\rightarrow Y
$$

be a function. Define a relation $R$ in $X$ given by

$$
R=\{(a,b):f(a)=f(b)\}.
$$

Examine whether $R$ is an equivalence relation or not.

Find the number of all one-one functions from the set

$$
A=\{1,2,3\}
$$

to itself.

Let

$$
A=\{1,2,3\}.
$$

Show that the number of relations containing $(1,2)$ and $(2,3)$ which are reflexive and transitive but not symmetric is three.

Show that the number of equivalence relations in the set

$$
\{1,2,3\}
$$

containing $(1,2)$ and $(2,1)$ is two.

Consider the identity function

$$
I_{\mathbb{N}}:\mathbb{N}\rightarrow\mathbb{N}
$$

defined as

$$
I_{\mathbb{N}}(x)=x
\qquad
\forall x\in\mathbb{N}.
$$

Show that although $I_{\mathbb{N}}$ is onto,

$$
I_{\mathbb{N}}+I_{\mathbb{N}}:
\mathbb{N}\rightarrow\mathbb{N}
$$

defined as

$$
(I_{\mathbb{N}}+I_{\mathbb{N}})(x)
=
I_{\mathbb{N}}(x)+I_{\mathbb{N}}(x)
=
x+x
=
2x
$$

is not onto.

Consider a function $$ f:\left[0,\frac{\pi}{2}\right]\rightarrow\mathbb{R} $$ given by $$ f(x)=\sin x $$ and $$ g:\left[0,\frac{\pi}{2}\right]\rightarrow\mathbb{R} $$ given by $$ g(x)=\cos x. $$
Show that $f$ and $g$ are one-one, but $f+g$ is not one-one.

Show that the function $f: \mathbf{R} \rightarrow\{x \in \mathbf{R}:-1<x<1\}$ defined by $f(x)=\frac{x}{1+|x|}$, $x \in \mathbf{R}$ is one one and onto function.

Show that the function $f: \mathbf{R} \rightarrow \mathbf{R}$ given by $f(x)=x^3$ is injective.

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

Find the number of all onto functions from the set $\{1,2,3, \ldots \ldots, n\}$ to itself.

Let $\mathrm{A}=\{-1,0,1,2\}, \mathrm{B}=\{-4,-2,0,2\}$ and $f, g: \mathrm{A} \rightarrow \mathrm{B}$ be functions defined by $f(x)=x^2-x, x \in \mathrm{~A}$ and $g(x)=2\left|x-\frac{1}{2}\right|-1, x \in \mathrm{~A}$. Are $f$ and $g$ equal? Justify your answer. (Hint: One may note that two functions $f: \mathrm{A} \rightarrow \mathrm{B}$ and $g: \mathrm{A} \rightarrow \mathrm{B}$ such that $f(a)=g(a) \forall a \in \mathrm{~A}$, are called equal functions).

Let $\mathrm{A}=\{1,2,3\}$. Then number of relations containing $(1,2)$ and $(1,3)$ which are reflexive and symmetric but not transitive is
(A) 1
(B) 2
(C) 3
(D) 4

Let $\mathrm{A}=\{1,2,3\}$. Then number of equivalence relations containing (1, 2) is
(A) 1
(B) 2
(C) 3
(D) 4