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.
▸ $Into$ : range is a $proper$ subset of the codomain, i.e. NOT onto; some target value is never 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 (needs

m \geq n

k = 0 \sum n (- 1)^{k}^{n} C_{k} (n - k)^{m}

, e.g. m elements onto 2:

2^{m} - 2

; 4 onto 3:

81 - 48 + 3 = 36

. Into

=

total

-

onto.

★ 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

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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.

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