Exercise 2.2 (Revised) - Chapter 2 - Relations & Functions - Ncert Solutions class 11 - Maths

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Chapter 2: Relations & Functions - NCERT Solutions for Class 11 Maths

Ex 2.2 Question 1.

Let $A=\{1,2,3$, $\qquad$ 14\}. Define a relation $\mathrm{R}$ from $\mathrm{A}$ to $\mathrm{A}$ by $\mathrm{R}=\{(x, y): 3 x-y=0$, where $x, y \in A\}$. Write down its domain co-domain and range.

Answer.

Given: $\mathrm{A}=\{1,2,3$, $\qquad$ $14\}$

The ordered pairs which satisfy $3 x-y=0$ are $(1,3),(2,6),(3,9)$ and $(4,12)$.
$
\begin{aligned}
& \therefore R=\{(1,3),(2,6),(3,9),(4,12)\} \\
& \text { Domain }=\{1,2,3,4\} \\
& \text { Range }=\{3,6,9,12\}
\end{aligned}
$
$
\text { Co-domain }=\{1,2,3 \text {, }
$
$\qquad$ $14\}$

Ex 2.2 Question 2.

Define a relation $\mathbf{R}$ on the set $\mathrm{N}$ of natural numbers $\mathbf{R}=\{(x, y): y=x+5, x$ is a natural number less than $4: x, y \in \mathrm{N}\}$. Depict this relationship using roster form. Write down the domain and the range.

Anwer.

Given: $\mathrm{R}=\{(x, y: y=x+5, x$ is a natural number less than $4: x, y \in \mathrm{N})\}$
Putting $x=1,2,3$ in $y=x+5$. we get $y=6,7,8$
$
\therefore R=\{(1,6),(2,7),(3,8)\}
$

Domain $=\{1,2,3\}$
$
\text { Range }=\{6,7,8\}
$
Ex 2.2 Question 3.

$A=\{1,2,35\}$ and $B=\{4,6,9\}$. Define a relation $R$ from $A$ to $B$ by $R=\{(x, y)$ : the

difference between $x$ and $y$ is odd: $x \in A, y \in B\}$. Write $\mathrm{R}$ in roster form.
Answer.

Given: $\mathrm{A}=\{1,2,3,5\}$ and $\mathrm{B}=\{4,6,9\}, x \in \mathrm{A}, y \in \mathrm{B}$
$
\begin{aligned}
& \therefore x-y=(1-4),(1-6),(1-9),(2-4),(2-6),(2-9),(3-4),(3-6)(3-9), \\
& (5-4),(5-6),(5-9) \\
& \Rightarrow x-y=-3,-5,-8,-2,-4=-7,-1,-3,-6,1=-1,-4 \\
& \therefore R=\{(1,4),(1,6),(2,9),(3,4),(3,6)(5,4),(5,6)\}
\end{aligned}
$
Ex 2.2 Question 4.

Figure shows a relationship between the sets $P$ and $Q$. Write this relation:
(i) in set-builder form
(ii) roster form

What is its domain and range?
Answer.

(i) Relation $\mathrm{R}$ in set-builder form is $\mathrm{R}=\{(x, y): y=x-2: x=5,6,7\}$
(ii) Relation $\mathrm{R}$ in roster form is $\mathrm{R}=\{(5,3),(6,4),(7,5)\}$

Domain $=\{5,6,7\}$
Range $=\{3,4,5\}$
Ex 2.2 Question 5.

Let $\mathrm{A}=\{1,2,3,4,6\}$. Let $\mathrm{R}$ be the relation on $\mathrm{A}$ defined by $\{(a, b): a, b \in \mathrm{A}, b$ is exactly divisible by $a\}$.
(i) Write $\mathrm{R}$ in roster form.

(ii) Find the domain of $R$.
(iii) Find the range of $R$.

Answer.

Given: $\mathrm{A}=\{1,2,3,4,6\}$

A set of ordered pairs $(a, b)$ where $b$ is exactly divisible by $a$.
(i) $\mathrm{R}=\{(1,1),(1,2),(1,3),(1,4),(1,6),(2,2),(2,4),(2,6),(3,3),(3,6),(4,4),(6,6)\}$
(ii) Domain of $\mathrm{R}=\{1,2,3,4,6\}$
(iii) Range of $\mathrm{R}=\{1,2,3,4,6\}$
Ex 2.2 Question 6.

Determine the domain and range of the relation $\mathrm{R}$ defined by
$
\mathbf{R}=\{(x, x+5): x \in(0,1,2,3,4,5)\}
$

Answer.

Given: $\mathrm{R}=\{(x, x+5): x \in(0,1,2,3,4,5)\}=\{(a, b): a=0,1,2,3,4,5\}$
$\therefore a=x$ and $b=x+5$
Putting $a=0,1,2,3,4,5$ we get $b=5,6,7,8,9,10$
$\therefore$ Domain of $\mathrm{R}=\{0,1,2,3,45\}$
Range of $\mathrm{R}=\{5,6,7,8,9,10\}$

Ex 2.2 Question 7.

Write the relation $\mathrm{R}=\left\{\left(x, x^3\right): x\right.$ is a prime number less than 10$\}$ in roster form.

Answer.

Given: $\mathrm{R}=\left\{\left(x, x^3\right): x\right.$ is a prime number less than 10$\}$
Putting $x=2,3,5,7$
$
R=\{(2,8),(3,27),(5,125),(7,343)\}
$
Ex 2.2 Question 8.

Let $A=\{x, y, z\}$ and $B=\{1,2\}$. Find the number of relations from $A$ to $B$.

Answer.

Given: $\mathrm{A}=\{x, y, z\}$ and $\mathrm{B}=\{1,2\}$
Number of elements in set $\mathrm{A}=3$ and Number of elements in set $\mathrm{B}=2$
$\therefore$ Number of subsets of $A \times B=3 \times 2=6$
Number of relations from A to $B=2^6$.
Ex 2.2 Question 9.

Let $R$ be the relation on $Z$ defined by $R=\{(a, b): a, b \in Z$ is an integer $\}$. Find the domain and range of $R$.

Answer.

Given: $\mathrm{R}=\{(a, b): a, b \in Z, a-b$ is an integer $\}$
$=\{(a, b): a, b \in Z$, both $a$ and $b$ are even or both $a$ and $b$ are odd $\}$
$=\{(a, b): a, b \in Z,(a$ and $b$ are even $) \cup(a$ and $b$ are odd $)\}$
$\therefore$ Domain of $\mathrm{R}=\mathrm{Z}$
Range of $\mathrm{R}=\mathrm{Z}$