10th Maths Chapter 1 Relations and Functions

10th Maths Chapter 1 Relations and Functions solutions, with the chapter's formulas and every exercise worked step by step, as per the 2026-27 Samacheer Kalvi textbook.

This chapter looks at ordered pairs, the Cartesian product of sets, and how a relation links one set to another. It then explains when a relation becomes a function, the ways a function can be shown, and the common types such as one-one, onto, many-one and into. In the exercises students form products, test relations, find domains and ranges, identify function types, and work out compositions of functions.

Exercises in this chapter

Key formulas

Cartesian product of A with B
$A \times B = \{(a,b) \mid \text{for all } a \in A,\ b \in B\}$
A relation R from A to B is a subset of A × B
$R \subseteq A \times B$
Condition for a relation R from X to Y to be a function
$\text{for every } x \in X \text{ there exists only one } y \in Y$
Identity function
$f(x) = x$
Reciprocal function
$f(x) = \dfrac{1}{x}$
Constant function
$f(x) = c$
Linear function
$f(x) = ax + b,\ a \neq 0$
Quadratic function
$f(x) = ax^{2} + bx + c,\ a \neq 0$
Cubic function
$f(x) = ax^{3} + bx^{2} + cx + d,\ a \neq 0$
Composition of two functions f : A → B and g : B → C
$g \circ f : A \to C,\quad (g \circ f)(x) = g(f(x)) \ \text{ for all } x \in A$
Composition is not commutative in general
$f \circ g \neq g \circ f$
Composition is associative
$f \circ (g \circ h) = (f \circ g) \circ h$

Chapter 1 one-mark practice

If $n(A \times B) = 6$ and $A = \{1,3\}$ then $n(B)$ is
1. (1) 1
2. (2) 2
3. (3) 3
4. (4) 6
Show answer
Answer: (3) 3
1. $n(A\times B)=n(A)\times n(B)$ .
2. $6=2\times n(B)$ .
3. $n(B)=3$ .
Question 1 of 10
$A = \{a,b,p\}$ , $B = \{2,3\}$ , $C = \{p,q,r,s\}$ then $n[(A \cup C) \times B]$ is
1. (1) 8
2. (2) 20
3. (3) 12
4. (4) 16
Show answer
Answer: (3) 12
1. $A\cup C=\{a,b,p,q,r,s\}$ , so $n(A\cup C)=6$ .
2. $n(B)=2$ .
3. $n[(A\cup C)\times B]=6\times 2=12$ .
Question 2 of 10
If $A = \{1,2\}$ , $B = \{1,2,3,4\}$ , $C = \{5,6\}$ and $D = \{5, 6, 7, 8\}$ then state which of the following statement is true.
1. (1) $(A \times C) \subset (B \times D)$
2. (2) $(B \times D) \subset (A \times C)$
3. (3) $(A \times B) \subset (A \times D)$
4. (4) $(D \times A) \subset (B \times A)$
Show answer
Answer: (1) $(A \times C) \subset (B \times D)$
1. $A=\{1,2\}\subset B=\{1,2,3,4\}$ and $C=\{5,6\}\subset D=\{5,6,7,8\}$ .
2. So every pair of $A\times C$ is also a pair of $B\times D$ .
3. Hence $(A\times C)\subset (B\times D)$ .
Question 3 of 10
If there are 1024 relations from a set $A = \{1, 2, 3, 4, 5\}$ to a set $B$ , then the number of elements in $B$ is
1. (1) 3
2. (2) 2
3. (3) 4
4. (4) 8
Show answer
Answer: (2) 2
1. Number of relations from $A$ to $B$ is $2^{n(A)\cdot n(B)}$ .
2. $2^{5\,n(B)}=1024=2^{10}$ .
3. $5\,n(B)=10\Rightarrow n(B)=2$ .
Question 4 of 10
The range of the relation $R = \{(x, x^2) \mid x$ is a prime number less than 13 $\}$ is
1. (1) $\{2,3,5,7\}$
2. (2) $\{2,3,5,7,11\}$
3. (3) $\{4,9,25,49,121\}$
4. (4) $\{1,4,9,25,49,121\}$
Show answer
Answer: (3) $\{4,9,25,49,121\}$
1. Primes less than 13 are $2,3,5,7,11$ .
2. Range consists of the squares $x^{2}$ .
3. Range $=\{4,9,25,49,121\}$ .
Question 5 of 10
If the ordered pairs $(a + 2, 4)$ and $(5, 2a + b)$ are equal then $(a,b)$ is
1. (1) $(2, -2)$
2. (2) $(5,1)$
3. (3) $(2,3)$
4. (4) $(3, -2)$
Show answer
Answer: (4) $(3, -2)$
1. Equal ordered pairs: $a+2=5$ and $2a+b=4$ .
2. $a=3$ .
3. $2(3)+b=4\Rightarrow b=-2$ .
Question 6 of 10
Let $n(A) = m$ and $n(B) = n$ then the total number of non-empty relations that can be defined from $A$ to $B$ is
1. (1) $m^n$
2. (2) $n^m$
3. (3) $2^{mn} - 1$
4. (4) $2^{mn}$
Show answer
Answer: (3) $2^{mn} - 1$
1. Total number of relations from $A$ to $B$ is $2^{mn}$ .
2. Remove the empty relation.
3. Non-empty relations $=2^{mn}-1$ .
Question 7 of 10
If $\{(a,8),(6,b)\}$ represents an identity function, then the value of $a$ and $b$ are respectively
1. (1) $(8,6)$
2. (2) $(8,8)$
3. (3) $(6,8)$
4. (4) $(6,6)$
Show answer
Answer: (1) $(8,6)$
1. For an identity function $f(x)=x$ , each pair is of the form $(x,x)$ .
2. $(a,8)\Rightarrow a=8$ .
3. $(6,b)\Rightarrow b=6$ .
Question 8 of 10
Let $A = \{1,2,3,4\}$ and $B = \{4,8,9,10\}$ . A function $f : A \to B$ given by $f = \{(1,4),(2,8),(3,9),(4,10)\}$ is a
1. (1) Many-one function
2. (2) Identity function
3. (3) One-to-one function
4. (4) Into function
Show answer
Answer: (3) One-to-one function
1. Images are $4,8,9,10$ — all different.
2. Different elements of $A$ have different images.
3. So $f$ is one-to-one.
Question 9 of 10
If $f(x) = 2x^2$ and $g(x) = \dfrac{1}{3x}$ , then $f \circ g$ is
1. (1) $\dfrac{3}{2x^2}$
2. (2) $\dfrac{2}{3x^2}$
3. (3) $\dfrac{2}{9x^2}$
4. (4) $\dfrac{1}{6x^2}$
Show answer
Answer: (3) $\dfrac{2}{9x^2}$
1. $(f\circ g)(x)=f\big(g(x)\big)=f\left(\dfrac{1}{3x}\right)$ .
2. $=2\left(\dfrac{1}{3x}\right)^{2}$ .
3. $=\dfrac{2}{9x^{2}}$ .
Question 10 of 10

Spotted a mistake? Email us at [email protected] and we will fix it.