Relations and Functions for CUET Maths

Q: What is a relation on a set?

A relation R on a set A is any subset of A × A. We write a R b when (a, b) R. The empty relation has no pairs at all; the universal relation is the whole of A × A. A set with n elements has n^2 ordered pairs, so it carries 2^n^2 relations.

Q: How do you check whether a relation is reflexive, symmetric or transitive?

Reflexive: (a, a) R for every a in A. One missing (a, a) is enough to fail. Symmetric: whenever (a, b) R, also (b, a) R. Transitive: whenever (a, b) and (b, c) are in R, so is (a, c).

Q: What is an equivalence class?

If R is an equivalence relation on A, the equivalence class of a is the set of everything related to a. Different classes never overlap, and together they cover A: the classes split A into separate pieces.

Q: When is a function one-one?

f: A → B is one-one (injective) when different inputs always give different outputs: f(a₁) = f(a₂) forces a₁ = a₂. To show it, start from f(a₁) = f(a₂) and solve; to disprove it, find two inputs with the same output.

Q: When is a function onto, and what is a bijection?

f: A → B is onto (surjective) when every element of B is the image of something: the range is the whole codomain. A function that is both one-one and onto is a bijection. The codomain matters: x^2 is onto [0, ∞) but not onto R.

About 5 min to learn · 125 questions

Relations and functions open Section B1 of CUET Maths, the section you answer if you take Mathematics. CUET asks the same few things here every year: which properties a relation has, what an equivalence class is, and whether a function is one-one or onto. Learn each rule here, from reflexive, symmetric and transitive relations and equivalence classes to one-one, onto and bijective functions and counting them, with worked examples, time-saving shortcuts and the traps CUET sets, then practise on free questions in CUET's own formats, each fully solved.

Syllabus area: Section B1: Mathematics

On this page · 10 sections

01What a relation is
02Reflexive, symmetric and transitive relations
03Equivalence relations and classes
04One-one functions
05Onto and bijective functions
06How CUET asks this chapter
07Exam shortcuts
08Common mistakes
09Quick revision
10Check your understanding

What a relation is

Take the six students of a study group and pair up any two who were born in the same month. That pairing is a relation: a set of ordered pairs $(a, b)$ taken from $A \times A$ . It has three properties worth checking. Everyone is born in the same month as themselves. If Asha shares a month with Ravi, Ravi shares it with Asha. And if Asha shares with Ravi and Ravi with Meera, then Asha shares with Meera. A relation with all three is an equivalence relation, and it splits the group into classes, one per birth month.

What is a relation on a set?

A relation $R$ on a set $A$ is any subset of $A \times A$ . We write $a \mathrel{R} b$ when $(a, b) \in R$ . The empty relation has no pairs at all; the universal relation is the whole of $A \times A$ . A set with $n$ elements has $n^{2}$ ordered pairs, so it carries $2^{n^{2}}$ relations.

Reflexive, symmetric and transitive relations

How do you check whether a relation is reflexive, symmetric or transitive?

Reflexive: $(a, a) \in R$ for every $a$ in $A$ . One missing $(a, a)$ is enough to fail.
Symmetric: whenever $(a, b) \in R$ , also $(b, a) \in R$ .
Transitive: whenever $(a, b)$ and $(b, c)$ are in $R$ , so is $(a, c)$ .

Symmetry and transitivity only make demands of pairs already in $R$ . That is why the empty relation is symmetric and transitive, though it is not reflexive.

Example 1 · A relation on a three-element set

On $A = \{2, 4, 6\}$ , $R = \{(2, 2), (2, 4), (4, 2)\}$ . Which properties does $R$ have?

Reflexive? $(4, 4)$ and $(6, 6)$ are missing, so no.
Symmetric? The only pair with a partner is $(2, 4)$ , and $(4, 2)$ is there. Yes.
Transitive? $(4, 2)$ and $(2, 4)$ are in $R$ , so $(4, 4)$ must be too. It is not, so no.

Answer

$R$ is symmetric only.

Watch out

Test transitivity on every chain, including the ones that turn back on themselves, such as $(4, 2)$ followed by $(2, 4)$ . Those chains demand a pair $(a, a)$ , and they are where most slips happen.

For a relation given by a rule on an infinite set, argue from the rule. " $a \le b$ " on $\mathbb{R}$ is reflexive and transitive but not symmetric, since $1 \le 2$ while $2 \le 1$ is false. " $\lvert a - b \rvert \le 2$ " is reflexive and symmetric but not transitive: 0 and 2 are related, 2 and 4 are related, but 0 and 4 are not.

Equivalence relations and classes

What is an equivalence class?

If $R$ is an equivalence relation on $A$ , the equivalence class of $a$ is the set of everything related to $a$ . Different classes never overlap, and together they cover $A$ : the classes split $A$ into separate pieces.

Example 2 · Classes of remainders

On $\mathbb{Z}$ , $a \mathrel{R} b$ when $a - b$ is divisible by 7. Find the class of 3, and the number of classes.

$R$ is an equivalence relation: $a - a = 0$ is divisible by 7; if 7 divides $a - b$ it divides $b - a$ ; and if it divides $a - b$ and $b - c$ , it divides their sum $a - c$ .
The class of 3 is every integer that leaves remainder 3 on division by 7: $\{\ldots, -11, -4, 3, 10, 17, \ldots\}$ .

Answer

The classes are the seven remainder classes $[0], [1], \ldots, [6]$ .

Key idea

An equivalence relation on a finite set is the same thing as a way of splitting the set into groups. That gives a quick count: $\{1, 2, 3\}$ can be split in 5 ways, so it has 5 equivalence relations.

One-one functions

When is a function one-one?

$f : A \to B$ is one-one (injective) when different inputs always give different outputs: $f(a_1) = f(a_2)$ forces $a_1 = a_2$ . To show it, start from $f(a_1) = f(a_2)$ and solve; to disprove it, find two inputs with the same output.

Example 3 · One-one or not

Is $f : \mathbb{R} \to \mathbb{R}$ , $f(x) = 5 - 2x$ , one-one? Is $g : \mathbb{R} \to \mathbb{R}$ , $g(x) = x^{2} - 1$ , one-one?

$5 - 2a = 5 - 2b$ gives $a = b$ , so $f$ is one-one.
$g(-3) = g(3) = 8$ , so $g$ is not one-one.

Answer

$f$ is one-one; $g$ is not.

Onto and bijective functions

When is a function onto, and what is a bijection?

$f : A \to B$ is onto (surjective) when every element of $B$ is the image of something: the range is the whole codomain. A function that is both one-one and onto is a bijection. The codomain matters: $x^{2}$ is onto $[0, \infty)$ but not onto $\mathbb{R}$ .

Example 4 · The same rule on different sets

Classify $f(n) = n + 5$ as a map $\mathbb{Z} \to \mathbb{Z}$ , and as a map $\mathbb{N} \to \mathbb{N}$ .

On $\mathbb{Z}$ : one-one, and any integer $m$ is $f(m - 5)$ , so it is onto: a bijection.
On $\mathbb{N}$ : still one-one, but 1, 2, 3, 4 and 5 have no natural number 5 below them, so it is not onto.

Answer

A bijection of $\mathbb{Z}$ ; one-one but not onto on $\mathbb{N}$ .

Example 5 · Counting functions

$A$ has 2 elements and $B$ has 3. How many functions from $A$ to $B$ are there, and how many are one-one?

Each of the 2 elements has 3 choices: $3^{2} = 9$ functions.
For one-one, the second element must avoid the first one's image: $3 \times 2 = 6$ .

Answer

9 functions, 6 of them one-one, and none onto, since 2 images cannot cover 3 elements.

How CUET asks this chapter

Types of relations: a relation on a small set or by a rule, and which of the three properties it has; the empty and universal relations; how many relations of a kind a set carries.
Equivalence: whether a relation is one, its classes, how many classes or equivalence relations there are.
Functions: whether a given function is one-one, onto, both or neither, often the same rule on two different domains; how many functions, one-one functions or onto functions there are.
Formats: two statements, assertion and reason, match the list, and "which of these are true".

Exam shortcuts

Relations on an $n$ -element set: $2^{n^{2}}$ ; reflexive ones: $2^{n^{2} - n}$ ; symmetric ones: $2^{\frac{n(n + 1)}{2}}$ .
Functions from an $m$ -set to an $n$ -set: $n^{m}$ ; one-one: ${}^{n}P_{m}$ (none if $m > n$ ); onto a 2-element set: $2^{m} - 2$ .
A bijection between finite sets needs equal sizes; from a finite set to itself, one-one and onto come together.
" $a - b$ divisible by $n$ " on $\mathbb{Z}$ is always an equivalence relation with $n$ classes.

Common mistakes

Calling the empty relation non-symmetric, or calling it reflexive.
Missing the chain that turns back on itself, such as $(1, 2), (2, 1)$ , which needs $(1, 1)$ .
Judging one-one or onto from the rule alone, forgetting the domain and codomain.
Taking the range of $e^{x}$ or $x^{2}$ to be all of $\mathbb{R}$ .

Quick revision

Relations

Reflexive: every $(a, a)$ ; symmetric: reverses; transitive: chains
Equivalence: all three; its classes split the set

Functions

One-one: equal outputs force equal inputs
Onto: range = codomain; bijection: both

Counting

$n^{m}$ functions, ${}^{n}P_{m}$ one-one
$2^{n^{2}}$ relations on $n$ elements

Next in Section B1, functions that undo the trigonometric ones: inverse trigonometric functions.

Next · 09Quick revision

Check your understanding

125 questions, grouped by type, easy to hard. Pick an answer and check it. The worked solution opens when you ask for it.

Types of relationsQ1 of 125Q1

Types of relations

45 questions

List I	List II
A. $\{(1, 1), (2, 2), (3, 3)\}$	I. Transitive only
B. $\{(1, 2), (2, 1)\}$	II. An equivalence relation
C. $\{(1, 2), (2, 3), (1, 3)\}$	III. Reflexive and transitive, but not symmetric
D. $\{(1, 1), (2, 2), (3, 3), (1, 2)\}$	IV. Symmetric only

List I	List II
A. $f : \mathbb{R} \to \mathbb{R}$ , $f(x) = 2x + 1$	I. One-one but not onto
B. $f : \mathbb{N} \to \mathbb{N}$ , $f(n) = 3n$	II. Neither one-one nor onto
C. $f : \mathbb{R} \to [0, \infty)$ , $f(x) = \lvert x \rvert$	III. One-one and onto
D. $f : \mathbb{R} \to \mathbb{R}$ , $f(x) = x^{2} + 1$	IV. Onto but not one-one

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