Sets, Relations and Functions Formula Sheet — JEE Main Mathematics

Every key Sets, Relations and Functions formula, definition and theorem for JEE Main Mathematics in one place — with common examiner traps and worked examples. Free to read; blurt from memory, then check your gaps.

Syllabus — topics coveredNTA · 11 sub-topics

Sets and their representation
Union, intersection and complement of sets
Venn diagrams
De Morgan's laws
Cartesian product
Relations
Types of relations
Equivalence relations
Functions
One-one, into and onto functions
Composition of functions

Sets — Representation, Types & Subsets

Set: A

well-defined collection

of distinct objects (elements). We write

a \in A

('a belongs to A') or

a \in / A

. Sets underpin relations, functions and almost every other topic — the whole unit builds on them.

Two ways to write a set

▸ $Roster$ : list the elements — ${2, 4, 6}$ (order & repetition don't matter).
▸ $Set-builder$ : state the rule — ${x : x is even, x < 7}$ .
▸Number sets: $N \subset Z \subset Q \subset R$ .
▸Intervals on $R$ : [a,b] closed, (a,b) open, [a,b) half-open.

Types of sets

Type	Meaning
$Empty / null$	$ϕ$ or ${}$ — has no elements
$Singleton$	exactly one element
$Finite / infinite$	the listing ends vs never ends
$Equal$ $(A = B)$	exactly the same elements
$Subset$ $A \subseteq B$	every $x \in A$ is also in B
$Power set$ P(A)	set of all subsets; $n (P (A)) = 2^{n}$
$Universal$ U	contains all sets under discussion
$Proper subset$ $A \subset B$	$A \subseteq B$ but $A \neq = B$

🎯 Exam · The power set and 2ⁿ

The

power set

P(A) is the set of

all subsets

of A. If

n (A) = n

, then

n (P (A)) = 2^{n}

(and the number of proper subsets is

2^{n} - 1

). Every set is a subset of itself, and

ϕ

is a subset of every set.

★ Remember · Equal sets vs equivalent sets

Equal

(

A = B

): exactly the same elements.

Equivalent

: the same

number

of elements (

n (A) = n (B)

). Equal

\Rightarrow

equivalent, but not the reverse.

⚠️ Watch out · ∈ vs ⊆, and φ vs {φ}

Use

\in

between an

element

and a set,

\subseteq

between two

sets

2 \in {1, 2}

but

{2} \subseteq {1, 2}

. Also

ϕ

is the empty set, while

{ϕ}

is a set containing one element — they are different.

🚫 Examiner Trap · Examiner traps

(1)

\emptyset

is a subset of EVERY set, and

\emptyset \neq = {\emptyset}

(one is empty, the other has ONE element). (2) Element vs set:

a \in A

but

{a} \subseteq A

— never mix

\in

and

\subseteq

. (3) Power set P(A) has

2^{n}

subsets,

2^{n} - 1

PROPER subsets,

2^{n} - 1

non-empty subsets. (4) In roster form order and repetition DON'T matter:

{1, 2, 2} = {1, 2}

Set Operations, Venn Diagrams & Counting

Venn diagrams for union, intersection, difference and complement, De Morgan's laws, and the inclusion-exclusion counting formulas for two and three sets — The four operations as Venn diagrams, De Morgan, and counting.

The four operations

▸ $A \cup B = {x : x \in A or x \in B}$ (in either).
▸ $A \cap B = {x : x \in A and x \in B}$ (in both).
▸ $A - B = {x : x \in A, x \in / B}$ (in A only).
▸ $A^{'} = U - A$ (everything outside A); disjoint sets have $A \cap B = ϕ$ .

★ Remember · De Morgan's laws

(A \cup B)^{'} = A^{'} \cap B^{'}

and

(A \cap B)^{'} = A^{'} \cup B^{'}

— the complement flips

\cup \leftrightarrow \cap

. With distributivity

A \cap (B \cup C) = (A \cap B) \cup (A \cap C)

, these simplify almost any set expression.

🎯 Exam · Inclusion–exclusion (counting)

n (A \cup B) = n (A) + n (B) - n (A \cap B)

. For three sets,

n (A \cup B \cup C) = n (A) + n (B) + n (C) - n (A \cap B) - n (B \cap C) - n (C \cap A) + n (A \cap B \cap C)

🎯 Exam · 'Exactly' vs 'at least' regions

Elements

only in

= n (A) - n (A \cap B)

. 'Exactly one of A,B'

= n (A) + n (B) - 2 n (A \cap B)

. Draw the Venn diagram and fill the innermost region first, then work outwards.

⚠️ Watch out · Don't double-count the overlap

The whole point of subtracting

n (A \cap B)

is that the overlap was counted in both n(A) and n(B). Forgetting it is the single most common counting error in this topic.

🚫 Examiner Trap · Examiner traps

(1) De Morgan FLIPS the operation:

(A \cup B)^{'} = A^{'} \cap B^{'}

and

(A \cap B)^{'} = A^{'} \cup B^{'}

. (2)

A - B = A \cap B^{'} \neq = B - A

(difference is NOT commutative). (3) For 3 sets you must ADD BACK the triple overlap:

n (A \cup B \cup C) = Σ n (A) - Σ n (A \cap B) + n (A \cap B \cap C)

. (4) 'Only in A'

= n (A) - n (A \cap B)

, not n(A); 'exactly one of A,B'

= n (A) + n (B) - 2 n (A \cap B)

Relations — Types & Equivalence Relations

Relation: A relation R from A to B is any

subset

of the cartesian product

A \times B = {(a, b) : a \in A, b \in B}

. Its

domain

is the set of first coordinates and its

range

the set of second coordinates.

The three building-block properties

▸ $Reflexive$ : $(a, a) \in R$ for every $a \in A$ .
▸ $Symmetric$ : $(a, b) \in R \Rightarrow (b, a) \in R$ .
▸ $Transitive$ : $(a, b), (b, c) \in R \Rightarrow (a, c) \in R$ .
▸ $Equivalence$ = reflexive AND symmetric AND transitive.

Types of a relation on

A

Relation	Condition
$Reflexive$	$(a, a) \in R$ for every a
$Symmetric$	$(a, b) \in R \Rightarrow (b, a) \in R$
$Transitive$	$(a, b), (b, c) \in R \Rightarrow (a, c) \in R$
$Antisymmetric$	$(a, b), (b, a) \in R \Rightarrow a = b$
$Equivalence$	reflexive AND symmetric AND transitive
$Identity / void$	${(a, a)}$ / empty $ϕ$ ; universal $= A \times A$

🎯 Exam · Counting relations

n (A \times B) = n (A) n (B) = pq

. A relation is any subset of

A \times A

, so a set with n elements has

2^{n^{2}}

relations on it. (Reflexive relations:

2^{n^{2} - n}

★ Remember · Equivalence ⇒ a partition

An equivalence relation splits A into disjoint

equivalence classes

— every element lies in exactly one class, and elements are related iff they share a class (e.g. 'congruent mod n' partitions

Z

into n classes).

⚠️ Watch out · Test each property with a counter-example

To disprove a property, one counter-example is enough: '

\geq

' on

R

is reflexive and transitive but

not

symmetric (

3 \geq 2

but

2 \neq \geq 3

), so it is not an equivalence relation. Antisymmetric

\neq =

'not symmetric'.

🚫 Examiner Trap · Examiner traps

(1) Reflexive needs (a,a) for EVERY a — one missing pair kills it; the EMPTY relation is symmetric & transitive but NOT reflexive (for non-empty A). (2) Symmetric and antisymmetric are NOT opposites — a relation can be both (e.g. equality). (3) Equivalence

=

reflexive AND symmetric AND transitive (all three) and partitions A into disjoint classes. (4) Count: relations on n-set

= 2^{n^{2}}

; reflexive ones

= 2^{n^{2} - n}

Functions — Definition, Domain, Range & Standard Functions

Function: A function

f : A \to B

assigns to

every

element of A

exactly one

element of B. So no input is left out and no input has two outputs — on a graph, every vertical line meets the curve at most once (vertical-line test).

Domain, codomain, range

▸ $Domain$ = all valid inputs (A); $codomain$ = target set B.
▸ $Range$ = the actual outputs, and range $\subseteq$ codomain.
▸To find a domain, exclude division by $0$ , even roots of negatives, and $lo g$ of $\leq 0$ .
▸Range often needs the inverse, completing the square, or a graph.

Standard real functions

Function	Rule — domain $\to$ range
$Identity$	$f (x) = x$ ; $R \to R$
$Constant$	$f (x) = c$ ; range ${c}$
$Polynomial$	$a_{0} + a_{1} x + \dots + a_{n} x^{n}$ ; domain $R$
$Rational$	$\frac{p ( x )}{q ( x )}$ , $q (x) \neq = 0$
$Modulus$	$∣ x ∣$ ; range $[0, \infty)$
$Exponential$	$a^{x}$ $(a > 0)$ ; range $(0, \infty)$
$Logarithm$	$lo g_{a} x$ ; domain $(0, \infty)$
$Square root$	$x$ ; domain $[0, \infty)$

🎯 Exam · Domains worth memorising

\frac{1}{x}

R - {0}

x

[0, \infty)

lo g_{a} x

(0, \infty)

a^{x}

: domain

R

, range

(0, \infty)

∣ x ∣

: domain

R

, range

[0, \infty)

★ Remember · Every function is a relation

A function is a special relation (a subset of

A \times B

) in which

no first coordinate repeats

. So all functions are relations, but most relations are not functions.

⚠️ Watch out · Codomain ≠ range

f : R \to R

f (x) = x^{2}

has codomain

R

but range

[0, \infty)

. Whether a function is 'onto' depends on the

codomain

you were given — read it carefully.

🚫 Examiner Trap · Examiner traps

(1) A function maps EVERY domain element to EXACTLY ONE output (vertical-line test cuts once) — one input with two outputs is NOT a function. (2) Domain rules: denominator

\neq = 0

, even root

\geq 0

lo g

argument

> 0

. (3) Range

\subseteq

codomain (range

=

ACTUAL outputs, codomain

=

target set). (4)

means the NON-negative root; range of

x

[0, \infty)

, not all reals.

Types of Functions — One-One, Onto & Bijective

Arrow diagrams of one-one (injective), many-one and onto (surjective) functions, how to test each, the bijective condition, and formulas for counting functions, injections and bijections — One-one, many-one and onto with tests, and counting formulas.

The three properties

▸ $One-one$ (injective): $f (x_{1}) = f (x_{2}) \Rightarrow x_{1} = x_{2}$ — distinct inputs, distinct outputs.
▸ $Onto$ (surjective): range $=$ codomain — every target value is hit.
▸ $Bijective$ = one-one AND onto.
▸Graph test: one-one $\Leftrightarrow$ every horizontal line meets the graph at most once.

🎯 Exam · Counting (|A| = m, |B| = n)

Total functions

A \to B

n^{m}

. One-one (needs

m \leq n

n (n - 1) \dots (n - m + 1) =^{n} P_{m}

. Bijections (need

m = n

): n!. Onto: use inclusion–exclusion.

★ Remember · Monotonic ⇒ one-one

A strictly increasing or strictly decreasing function is automatically one-one. Checking the sign of

f^{'} (x)

is often the quickest injectivity test for a differentiable function.

🎯 Exam · Bijective ⇔ invertible

A function has an inverse if and only if it is

bijective

. So 'find

f^{- 1}

' problems first need you to confirm (or arrange the codomain so) that f is one-one and onto.

⚠️ Watch out · Onto depends on the stated codomain

f (x) = x^{2}

is onto as

f : R \to [0, \infty)

but

not

onto as

f : R \to R

. Always check injectivity and surjectivity against the codomain you are actually given.

🚫 Examiner Trap · Examiner traps

(1) One-one test:

f (x_{1}) = f (x_{2}) \Rightarrow x_{1} = x_{2}

(horizontal line cuts once); onto test: range

=

codomain. (2) A function may be NEITHER one-one nor onto — the four types aren't exhaustive of one property. (3) Bijective

=

one-one AND onto

\Rightarrow

invertible. (4) Counts (

∣ A ∣ = m, ∣ B ∣ = n

): one-one needs

m \leq n

giving

\frac{n !}{( n - m )!}

; bijections (

m = n

)

= n!

; total functions

= n^{m}

Composition & Inverse of Functions

Mapping diagram: set A maps by f to set B, then by g to set C; the composite g of f maps A directly to C — Apply f first, then g: $g \circ f$ maps A straight to C.

Composition g∘f:

(g \circ f) (x) = g (f (x))

— apply f first, then g. It is defined only when the range of f lies inside the domain of g. Composition is

associative

but

not commutative

: usually

g \circ f \neq = f \circ g

Composition facts

▸Order matters: $g \circ f$ means do f first, then g.
▸one-one $\circ$ one-one $=$ one-one; onto $\circ$ onto $=$ onto.
▸So bijective $\circ$ bijective $=$ bijective (still invertible).

Inverse function f⁻¹

▸Exists $\Leftrightarrow$ f is $bijective$ .
▸ $f (a) = b \Leftrightarrow f^{- 1} (b) = a$ .
▸ $f^{- 1} \circ f = I_{A}$ and $f \circ f^{- 1} = I_{B}$ .
▸Graph of $f^{- 1}$ = reflection of f in the line $y = x$ .

🎯 Exam · Reverse the order when inverting a composite

(g \circ f)^{- 1} = f^{- 1} \circ g^{- 1}

— like taking off shoes then socks. The same reversal holds for the composite being one-one/onto:

g \circ f

one-one

\Rightarrow f

is one-one.

★ Remember · How to find f⁻¹

Write

y = f (x)

, solve for x in terms of y, then swap symbols to get

f^{- 1} (x)

. The domain of

f^{- 1}

is the range of f, and its range is the domain of f.

⚠️ Watch out · f⁻¹ is not 1/f

The inverse function

f^{- 1}

not

the reciprocal

\frac{1}{f ( x )}

. They are completely different objects —

f^{- 1}

undoes f, whereas

1/ f

is just division.

🚫 Examiner Trap · Examiner traps

(1)

(g \circ f) (x) = g (f (x))

— apply f FIRST, then g (read right-to-left). (2) Composition is NOT commutative:

g \circ f \neq = f \circ g

in general. (3) Inverse REVERSES order (socks-shoes):

(g \circ f)^{- 1} = f^{- 1} \circ g^{- 1}

. (4)

f^{- 1}

exists IFF f is bijective, and the graph of

f^{- 1}

is the reflection of f in the line

y = x

Special Functions & Their Graphs

Graphs of the modulus, greatest-integer, signum and identity functions, plus a summary of even/odd/periodic functions and the step/special functions — The key graphs, and even/odd/periodic behaviour.

Graphs to recognise

▸ $Modulus$ $∣ x ∣$ : V-shape, range $[0, \infty)$ .
▸ $Greatest integer$ [x]: largest integer $\leq x$ — a rising staircase.
▸ $Fractional part$ ${x} = x - [x]$ : range $[0, 1)$ , period $1$ .
▸ $Signum$ : $- 1, 0, + 1$ for $x < 0, x = 0, x > 0$ ; $identity$ $y = x$ .

★ Remember · Even and odd by symmetry

Even

f (- x) = f (x)

— graph symmetric about the y-axis (

x^{2}

cos x

∣ x ∣

Odd

f (- x) = - f (x)

— symmetric about the origin (

x^{3}

sin x

). Most functions are neither.

🎯 Exam · Periods worth knowing

A function is

periodic

f (x + T) = f (x)

; the least positive T is the period.

sin x, cos x

2 π

;

tan x, cot x

π

;

∣ sin x ∣

π

;

{x}

1

; a constant has no fundamental period.

★ Remember · Any function splits into even + odd

Every f can be written as

f (x) = even \frac{f ( x ) + f ( - x )}{2} + odd \frac{f ( x ) - f ( - x )}{2}

— a handy trick for symmetry and integration problems.

⚠️ Watch out · [x] is not rounding

[x] is the

floor

, not rounding:

[2.7] = 2

but also

[- 2.3] = - 3

(the greatest integer

not exceeding

x, so it steps down for negatives). And

[x] + [- x] = 0

only when x is an integer, else

- 1

🚫 Examiner Trap · Examiner traps

(1) Greatest integer goes DOWN for negatives:

[- 2.3] = - 3

(not

- 2

);

[x] \leq x < [x] + 1

. (2) Fractional part

{x} = x - [x] \in [0, 1)

is ALWAYS non-negative, even for negative x (

{- 2.3} = 0.7

). (3)

sgn (0) = 0

(not

\pm 1

). (4)

∣ x ∣

has range

[0, \infty)

, is even, and is continuous everywhere but NOT differentiable at

x = 0

More JEE Main Mathematics formula sheets

Frequently Asked Questions

What are the most important Sets, Relations and Functions formulas for JEE Main?

This Sets, Relations and Functions formula sheet covers all the high-yield Mathematics formulas, definitions and theorems you need for JEE Main, across Sets and their representation, Union, intersection and complement of sets, Venn diagrams, De Morgan's laws, Cartesian product — each shown with the key result and, where useful, a worked example.

Is this Sets, Relations and Functions formula sheet free?

Yes — the full chapter formula sheet is free to read online, no login or payment required.

How should I revise Sets, Relations and Functions formulas?

Blurt the Sets, Relations and Functions formulas from memory, then check against this sheet to find your gaps — and practise a few previous-year questions on the chapter to make sure you can apply them under time pressure.

Also useful: all formula sheets · JEE Main previous-year papers · most important chapters.