How much of JEE Main comes from Sets Relations and Functions?

JEEGenie estimates about 3% of JEE Main marks come from Sets Relations and Functions, based on recent papers. The share varies from year to year.

Which formulas matter most in Sets Relations and Functions?

The Sets Relations and Functions sheet on this page has 22 formula entries, starting with Subset and Superset, Power Set, Union and Intersection, Difference and Complement, De Morgan's Laws. Each one comes with when to use it and the trap to avoid.

What topics does Sets Relations and Functions cover for JEE?

Sets Relations and Functions covers Sets and Operations, Relations, Types of Functions, Domain and Range, Composite and Inverse Functions, Binary Operations.

How should I revise Sets Relations and Functions before the exam?

Learn each formula together with its condition of use, read the trap listed beside it, then practise questions and review every mistake. The practice link on this page opens Sets Relations and Functions questions on JEEGenie.

How many previous-year questions are there on Sets Relations and Functions?

JEEGenie has 541 previous-year questions on Sets Relations and Functions, and this page shows a few with worked explanations.

Is practice on Sets Relations and Functions free on JEEGenie?

Sign-up is free and includes a 7-day Pro trial. After the trial, free accounts keep a limited daily quota.

Sets Relations and Functions for JEE Main & Advanced — formulas, what is asked, and traps

JEEGenie estimates about 3% of JEE Main marks come from Sets Relations and Functions. This page covers 22 formula entries, each with when to use it, what JEE asks and the traps. A common slip: Confusing 'well-defined' with 'vague'—e.g., 'set of tall students' is not a set.

Key points to know

Set and Its Representations. A set is a well-defined collection of distinct objects. It can be represented in roster (list elements) or set-builder (describe property) form.
Subset and Superset. Set A is a subset of B ( $A \subseteq B$ ) if every element of A is in B. B is then a superset of A.
Power Set. The power set of a set A, denoted P(A), is the set of all subsets of A. Its cardinality is $2^{n}$ where n = |A|.
Union and Intersection. Union ( $A \cup B$ ) is set of elements in A or B or both. Intersection ( $A \cap B$ ) is set of elements in both A and B.
Difference and Complement. Difference ( $A - B$ ) is elements in A but not in B. Complement ( $A^{'}$ or $\overset{ˉ}{A}$ ) is elements not in A, relative to universal set U.
Ordered Pair and Cartesian Product. An ordered pair (a,b) has first element a and second b; order matters. Cartesian product $A \times B$ is set of all ordered pairs from A and B.
Relation as a Subset of Cartesian Product. A relation R from set A to set B is a subset of $A \times B$ . If (a,b) ∈ R, we write a R b.
Types of Relations: Reflexive, Symmetric, Transitive. A relation on a set A is reflexive if every element relates to itself; symmetric if aRb implies bRa; transitive if aRb and bRc imply aRc.
Equivalence Relation and Equivalence Class. An equivalence relation is reflexive, symmetric, and transitive. It partitions the set into equivalence classes of mutually related elements.
Function as a Special Relation. A function f from A to B is a relation where each element in A has exactly one image in B. Denoted $f : A \to B$ .
Domain, Codomain, and Range of a Function. Domain is set of inputs (A). Codomain is set of possible outputs (B). Range is set of actual outputs (subset of codomain).
One-One (Injective) Function. A function is one-one if distinct inputs give distinct outputs: $f (a) = f (b) ⟹ a = b$ .
Onto (Surjective) Function. A function is onto if every element of codomain has a preimage: Range = Codomain.
Bijective Function and Inverse. A function is bijective if it is both one-one and onto. Only bijective functions have inverses.
Standard Sets and Notations. N: natural numbers {1,2,3,...}, Z: integers, Q: rationals, R: reals, C: complex. Also, $\emptyset$ is empty set, U is universal set.
Venn Diagrams for Set Problems. Use overlapping circles to visualize unions, intersections, complements. Useful for word problems with multiple sets.

Formula sheet

Name	Formula	When to use
Subset and Superset	$A \subseteq B ⟺ \forall x \in A, x \in B$	When comparing sets or proving set relationships.
Power Set	$∣ P (A) ∣ = 2^{∣ A ∣}$	When counting subsets or dealing with all possible combinations of elements.
Union and Intersection	$A \cup B = {x : x \in A or x \in B}$ , $A \cap B = {x : x \in A and x \in B}$	When combining or finding common elements of sets.
Difference and Complement	$A - B = {x : x \in A and x \notin B}$ , $A^{'} = U - A$	When finding elements exclusive to one set or relative to a universe.
De Morgan's Laws	$(A \cup B)^{'} = A^{'} \cap B^{'}$ , $(A \cap B)^{'} = A^{'} \cup B^{'}$	When simplifying expressions with complements, unions, and intersections.
Cardinality of Union (Inclusion-Exclusion)	$∣ A \cup B ∣ = ∣ A ∣ + ∣ B ∣ - ∣ A \cap B ∣$ , $∣ A \cup B \cup C ∣ = ∣ A ∣ + ∣ B ∣ + ∣ C ∣ - ∣ A \cap B ∣ - ∣ B \cap C ∣ - ∣ C \cap A ∣ + ∣ A \cap B \cap C ∣$	When counting elements in unions, especially in word problems with overlapping categories.
Ordered Pair and Cartesian Product	$A \times B = {(a, b) : a \in A, b \in B}$ , $∣ A \times B ∣ = ∣ A ∣ \cdot ∣ B ∣$	When defining relations or functions, or in coordinate geometry.
Relation as a Subset of Cartesian Product	$R \subseteq A \times B$	When defining connections between elements of two sets.
Types of Relations: Reflexive, Symmetric, Transitive	Reflexive: $\forall a \in A, a R a$ ; Symmetric: $a R b ⟹ b R a$ ; Transitive: $a R b \land b R c ⟹ a R c$	When classifying relations as equivalence or partial order.
Equivalence Relation and Equivalence Class	Equivalence class of a: $[a] = {x \in A : x R a}$	When grouping elements that share a property, e.g., congruence modulo n.
Function as a Special Relation	$f \subseteq A \times B$ such that $\forall a \in A, \exists! b \in B$ with $(a, b) \in f$	When defining a mapping from one set to another.
Domain, Codomain, and Range of a Function	Domain = ${x : f (x) defined}$ , Range = ${f (x) : x \in Domain}$	When analyzing a function's behavior or solving equations.
One-One (Injective) Function	$f (a) = f (b) ⟹ a = b$	When determining if a function has an inverse or is invertible.
Onto (Surjective) Function	$\forall y \in B, \exists x \in A$ such that $f (x) = y$	When checking if a function covers all possible outputs.
Bijective Function and Inverse	$f^{- 1} (y) = x$ if $f (x) = y$ , for bijective $f$	When finding inverse functions or solving functional equations.
Composition of Functions	$(g \circ f) (x) = g (f (x))$	When combining functions or solving functional equations.
Finding Domain of a Function	For $f (x) = \frac{1}{g (x)}$ , $g (x) \neq 0$ ; for $\sqrt{g (x)}$ , $g (x) \geq 0$ ; for $\log (g (x))$ , $g (x) > 0$	When given a function expression and asked for its natural domain.
Finding Range of a Function	For quadratic $f (x) = a x^{2} + b x + c$ , range is $[f (- b / 2 a), \infty)$ if a>0, else $(- \infty, f (- b / 2 a)]$	When asked to find set of all possible outputs of a function.
Checking Injectivity and Surjectivity	If $f^{'} (x) > 0$ (or $< 0$ ) throughout an interval domain, then $f$ is strictly monotonic, hence injective.	When classifying functions or proving bijectivity.
Solving Functional Equations	For Cauchy equation $f (x + y) = f (x) + f (y)$ , if continuous, $f (x) = c x$ .	When given a functional equation and asked to find f(x) or its properties.
Standard Sets and Notations	$N \subset Z \subset Q \subset R \subset C$	When working with any set theory or function problems.
Venn Diagrams for Set Problems	For two sets, draw two overlapping circles; for three, three overlapping circles.	When solving problems involving multiple set relationships, especially with given cardinalities.

Open the Sets Relations and Functions formula page

What JEE asks

Set and Its Representations. JEE often asks to identify which collections are sets or to convert between roster and set-builder forms.
Subset and Superset. JEE tests subset relations in multiple-choice, often with power sets or Venn diagrams.
Power Set. JEE asks for number of elements in power set or to list subsets in problems.
Union and Intersection. JEE tests these in set operations problems, often with Venn diagrams or word problems.
Difference and Complement. JEE uses these in set identities and proofs, e.g., De Morgan's laws.
De Morgan's Laws. JEE often asks to prove or apply these laws in set theory problems.
Cardinality of Union (Inclusion-Exclusion). JEE frequently tests this in survey problems or Venn diagram questions.
Ordered Pair and Cartesian Product. JEE tests Cartesian product in relation and function definitions, and in counting problems.
Relation as a Subset of Cartesian Product. JEE asks to identify relations from given ordered pairs or to find domain/range.
Types of Relations: Reflexive, Symmetric, Transitive. JEE frequently tests checking these properties for given relations, often in matrix or set form.
Equivalence Relation and Equivalence Class. JEE tests equivalence relations in problems like 'check if R is equivalence' or find number of equivalence classes.
Function as a Special Relation. JEE tests function definition, domain, range, and whether a relation is a function.
Domain, Codomain, and Range of a Function. JEE often asks to find domain or range of given functions, especially with radicals or fractions.
One-One (Injective) Function. JEE tests injectivity using algebraic or graphical methods, often with trigonometric or polynomial functions.
Onto (Surjective) Function. JEE tests surjectivity in function classification problems, often with finite sets or polynomial ranges.
Bijective Function and Inverse. JEE tests bijectivity and inverse in problems like 'find f^{-1}' or 'prove f is bijective'.
Composition of Functions. JEE tests composition in problems like 'find f(g(x))' or 'if f and g are given, find fog'.
Finding Domain of a Function. JEE frequently asks domain questions with combinations of radicals, logs, and fractions.
Finding Range of a Function. JEE tests range in problems with trigonometric, exponential, or rational functions.
Checking Injectivity and Surjectivity. JEE often asks to prove a function is one-one/onto using algebraic manipulation or calculus.
Solving Functional Equations. JEE tests functional equations in advanced problems, often with linear or exponential forms.
Standard Sets and Notations. JEE uses these notations in domain/range problems and set operations.
Venn Diagrams for Set Problems. JEE uses Venn diagrams in survey problems and to find unknown set sizes.

Common traps

Confusing 'well-defined' with 'vague'—e.g., 'set of tall students' is not a set. Set and Its Representations
Thinking $\subset$ and $\subseteq$ are same— $\subset$ means proper subset (A ≠ B). Subset and Superset
Forgetting that empty set and A itself are always in P(A). Power Set
Using 'or' in union as exclusive—it is inclusive. Union and Intersection
Thinking $A - B = B - A$ —they are different unless A = B. Difference and Complement
Applying laws incorrectly—e.g., thinking $(A \cup B)^{'} = A^{'} \cup B^{'}$ . De Morgan's Laws
Forgetting to subtract intersection in two-set case. Cardinality of Union (Inclusion-Exclusion)
Thinking (a,b) = (b,a)—order matters unless a=b. Ordered Pair and Cartesian Product
Thinking every subset of $A \times B$ is a function—it's a relation, not necessarily a function. Relation as a Subset of Cartesian Product
Assuming a relation is reflexive if it has some self-pairs—it must have all. Types of Relations: Reflexive, Symmetric, Transitive
Thinking an equivalence relation must be antisymmetric—it's not. Equivalence Relation and Equivalence Class
Thinking a function can have one input with multiple outputs—it cannot. Function as a Special Relation
Thinking range equals codomain—range is subset, not necessarily equal. Domain, Codomain, and Range of a Function
Judging one-one-ness without fixing the domain; e.g. $x^{2}$ is one-one on $[0, \infty)$ but not on $R$ . One-One (Injective) Function

Topics in Sets Relations and Functions

Sets and Operations
Relations
Types of Functions
Domain and Range
Composite and Inverse Functions
Binary Operations

Previous-year questions on Sets Relations and Functions

JEEGenie has 541 previous-year questions on Sets Relations and Functions. A few worked examples follow.

Year	Questions
2026	37
2025	47
2024	51
2023	54
2022	57
2021	58
2020	25
2019	52
2018	13
2017	8
2016	12
2015	5
2014	6
2013	7
2012	7
2011	4
2010	1
2009	3
2008	4
2007	6
2006	2
2005	6
2004	8
2003	7
2002	7
2001	9
2000	3
1999	2
1998	2
1996	5
1994	2
1992	2
1991	1
1989	2
1988	1
1986	2
1985	2
1984	3
1983	9
1982	1
1981	2
1980	2
1979	1
1978	3

Question 1 medium

Let

S = {x \in [- π, π] : \sin x (\sin x + \cos x) = a, a \in Z}

. Then

n (S)

is equal to:

Answer: D

Explanation.

\sin

\sin

\cos

x) =

\sin^{2}

x +

\sin

\cos

x = (1-

\cos

2x)/2 + (

\sin

2x)/2 = [1 +

\sin

2x -

\cos

2x]/2. So

\sin

\sin

\cos

x)=a becomes 1+

\sin

2x-

\cos

2x=2a, i.e.

\sin

2x-

\cos

2x=2a-1. Writing

\sin

2x-

\cos

2x = sqrt(2)

\sin

(2x-pi/4), we get

\sin

(2x-pi/4) = (2a-1)/sqrt(2). For a solution to exist the right side must lie in [-1,1]: -sqrt(2) <= 2a-1 <= sqrt(2), i.e. (1-sqrt(2))/2 <= a <= (1+sqrt(2))/2, approximately -0.207 <= a <= 1.207. Since a is an integer, a=0 or a=1 only. For x in [-pi,pi], let

θ

=2x-pi/4; then

θ

ranges over [-9pi/4, 7pi/4], an interval of total length 4

π

(two full periods). Case a=0:

\sin

(

θ

)=-1/sqrt(2). Solutions are

θ

=-pi/4+2pi*n or

θ

=5pi/4+2pi*n. Within [-9pi/4,7pi/4]: from

θ

=-pi/4+2pi*n we get

θ

=-9pi/4, -pi/4, 7pi/4 (3 values, including both endpoints of the domain); from

θ

=5pi/4+2pi*n we get

θ

=-3pi/4, 5pi/4 (2 values). That gives 5 solutions for a=0. Case a=1:

\sin

(

θ

)=1/sqrt(2). Solutions are

θ

=pi/4+2pi*n or

θ

=3pi/4+2pi*n. Within [-9pi/4,7pi/4]: from

θ

=pi/4+2pi*n we get

θ

=-7pi/4, pi/4 (2 values); from

θ

=3pi/4+2pi*n we get

θ

=-5pi/4, 3pi/4 (2 values). That gives 4 solutions for a=1. Total number of solutions in [-pi,pi]: 5 + 4 = 9. Answer: (D)

From NTA/IIT papers: JEE Main 2026 · 2 Apr Shift 1

Question 2 easy

θ \in [- 2 π, 2 π]

, then the number of solutions of

2 \sqrt{2} \cos^{2} θ + (2 - \sqrt{6}) \cos θ - \sqrt{3} = 0

, is equal to:

Answer: A

Explanation. We solve the quadratic in

\cos θ

2 \sqrt{2} \cos^{2} θ + (2 - \sqrt{6}) \cos θ - \sqrt{3} = 0.

Discriminant:

Δ = (2 - \sqrt{6})^{2} + 8 \sqrt{2} \sqrt{3} = 4 + 6 - 4 \sqrt{6} + 8 \sqrt{6} = 10 + 4 \sqrt{6} .

Note

10 + 4 \sqrt{6} = (\sqrt{6} + 2)^{2}

. So

\cos θ = \frac{- (2 - \sqrt{6}) \pm (\sqrt{6} + 2)}{4 \sqrt{2}} .

First root:

- (2 - \sqrt{6}) + (\sqrt{6} + 2) = 2 \sqrt{6}

, so

\cos θ = \frac{2 \sqrt{6}}{4 \sqrt{2}} = \frac{\sqrt{3}}{2}

. Second root:

- (2 - \sqrt{6}) - (\sqrt{6} + 2) = - 4

, so

\cos θ = \frac{- 4}{4 \sqrt{2}} = - \frac{1}{\sqrt{2}}

. Thus

\cos θ = \frac{\sqrt{3}}{2}

\cos θ = - \frac{1}{\sqrt{2}}

. In

[- 2 π, 2 π]

, each cosine value occurs 4

\times

(two full periods). Total solutions:

4 + 4 = 8

. The answer is option A.

From NTA/IIT papers: JEE Main 2025 · 2 Apr Shift 1

Question 3 hard

In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let

m

and

n

respectively be the least and the most number of students who studied all the three subjects. Then

m + n

is equal to ___________.

Answer: 45

Explanation. 210 of the 220 students study at least one of the three subjects, since 10 study none. Let

T

be the number who study all three. By inclusion-exclusion:

210 = M + P + C - (40 + 50 + 30) + T \Rightarrow T = 330 - (M + P + C) .

**Least value.**

M + P + C

is largest at

130 + 95 + 90 = 315

, which gives

T = 15

. Check that every region is non-negative: - only Maths:

130 - 90 + 15 = 55

- only Physics:

95 - 70 + 15 = 40

- only Chemistry:

90 - 80 + 15 = 25

- the pairwise-only regions:

25

35

15

All are valid, so

m = 15

. **Greatest value.**

T

cannot exceed the smallest pairwise overlap, so

T \leq n (P \cap C) = 30

T = 30

needs

M + P + C = 300

, for example

M = 130

P = 95

C = 75

. The regions are then: - only Maths:

70

- only Physics:

55

- only Chemistry:

25

P \cap C

only:

0

, the other pairwise-only regions

10

and

20

All are valid, so

n = 30

m + n = 15 + 30 = 45

Answer: 45

From NTA/IIT papers: JEE Main 2024 · 4 Apr Shift 1

Question 4 medium

From the top

A

of a vertical wall

A B

of height

30 m

, the angles of depression of the top

P

and bottom

Q

of a vertical tower

P Q

are

15^{\circ}

and

60^{\circ}

respectively,

B

and

Q

are on the same horizontal level. If

C

is a point on

A B

such that

C B = P Q

, then the area (in

m^{2}

) of the quadrilateral

B C P Q

is equal to :

$200 (3 - \sqrt{3})$
$300 (\sqrt{3} - 1)$
$300 (\sqrt{3} + 1)$
$600 (\sqrt{3} - 1)$

Answer: D

Explanation. Let the wall be

A B

with

A B = 30 m

, and let

B Q

be horizontal. Let the horizontal distance between the wall and the tower be

B Q = x

. Since the angle of depression of the bottom

Q

from

A

60^{\circ}

, in right triangle

A B Q

\tan 60^{\circ} = \frac{A B}{B Q} = \frac{30}{x}

\sqrt{3} = \frac{30}{x}

x = \frac{30}{\sqrt{3}} = 10 \sqrt{3}

B Q = 10 \sqrt{3} m .

Let the height of the tower be

P Q = h

. The top

P

is vertically above

Q

, so the horizontal distance from

A

to the vertical line through

P

is also

B Q = 10 \sqrt{3}

. The angle of depression of

P

from

A

15^{\circ}

. The vertical difference between

A

and

P

30 - h

. Thus in the right triangle formed,

\tan 15^{\circ} = \frac{30 - h}{10 \sqrt{3}} .

Now

\tan 15^{\circ} = 2 - \sqrt{3} .

Therefore,

30 - h = 10 \sqrt{3} (2 - \sqrt{3})

30 - h = 20 \sqrt{3} - 30

h = 60 - 20 \sqrt{3}

h = 20 (3 - \sqrt{3}) .

P Q = 20 (3 - \sqrt{3}) m .

Now

C

is on

A B

such that

C B = P Q = 20 (3 - \sqrt{3}) .

Since

A B = 30

A C = A B - C B = 30 - 20 (3 - \sqrt{3})

A C = 30 - 60 + 20 \sqrt{3} = 20 \sqrt{3} - 30.

The quadrilateral

B C P Q

has

B C ∥ P Q

, both vertical, and

B Q

horizontal. Also

C P

is horizontal

∵

C

and

P

are at the same height. Hence

B C P Q

is a trapezium with parallel sides

B C

and

P Q

, and the perpendicular distance between them is

B Q

. Thus its area is

Area = \frac{1}{2} (B C + P Q) \cdot B Q .

Since

B C = P Q

Area = \frac{1}{2} (2 P Q) \cdot B Q = P Q \cdot B Q .

Therefore,

Area = 20 (3 - \sqrt{3}) \cdot 10 \sqrt{3}

= 200 \sqrt{3} (3 - \sqrt{3})

= 200 (3 \sqrt{3} - 3)

= 600 (\sqrt{3} - 1) .

Answer: (D)

From NTA/IIT papers: JEE Main 2023 · 6 Apr Shift 1

Frequently asked questions

How much of JEE Main comes from Sets Relations and Functions?: JEEGenie estimates about 3% of JEE Main marks come from Sets Relations and Functions, based on recent papers. The share varies from year to year.
Which formulas matter most in Sets Relations and Functions?: The Sets Relations and Functions sheet on this page has 22 formula entries, starting with Subset and Superset, Power Set, Union and Intersection, Difference and Complement, De Morgan's Laws. Each one comes with when to use it and the trap to avoid.
What topics does Sets Relations and Functions cover for JEE?: Sets Relations and Functions covers Sets and Operations, Relations, Types of Functions, Domain and Range, Composite and Inverse Functions, Binary Operations.
How should I revise Sets Relations and Functions before the exam?: Learn each formula together with its condition of use, read the trap listed beside it, then practise questions and review every mistake. The practice link on this page opens Sets Relations and Functions questions on JEEGenie.
How many previous-year questions are there on Sets Relations and Functions?: JEEGenie has 541 previous-year questions on Sets Relations and Functions, and this page shows a few with worked explanations.
Is practice on Sets Relations and Functions free on JEEGenie?: Sign-up is free and includes a 7-day Pro trial. After the trial, free accounts keep a limited daily quota.