Sets, Relations and Functions — JEE Main Mathematics Notes & Formulas

1. Chapter at a Glance

This chapter is the language of the whole of mathematics. Sets tell us how to group objects, relations tell us how objects of two sets are connected, and functions are special relations in which every input has exactly one output. In JEE Main, one or two questions come every shift — usually on counting relations or functions, equivalence relations, domain and range, and one-one/onto checking.

What JEE asks most

Number of relations / reflexive / symmetric relations on a finite set.
Checking whether a given relation is reflexive, symmetric, transitive or an equivalence relation.
Domain of functions involving √, log and inverse trigonometric functions.
Number of one-one and onto functions; composition and inverse of functions.

2. Sets — Basics and Operations

A set is a well-defined collection of objects. "Well-defined" means we can clearly say whether an object belongs to it or not. The set of "good students" is not a set, but the set of "students who scored above 90" is.

Types of sets

Empty set (∅): has no element. It is a subset of every set.
Finite and infinite sets: {1, 2, 3} is finite; ℕ is infinite.
Equal sets: same elements (order and repetition do not matter).
Subset: A ⊆ B if every element of A is in B. A set with n elements has 2ⁿ subsets, and 2ⁿ − 1 proper subsets.
Power set P(A): the set of all subsets of A; n(P(A)) = 2^n(A).
Universal set (U): the set containing all objects under discussion.

Operations

Union A ∪ B = {x : x ∈ A or x ∈ B}; Intersection A ∩ B = {x : x ∈ A and x ∈ B}.
Difference A − B = {x : x ∈ A, x ∉ B} = A ∩ B′; Complement A′ = U − A.
Symmetric difference A Δ B = (A − B) ∪ (B − A).
De Morgan's laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.
Distributive laws: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C); A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

Venn diagram: the three regions A − B, A ∩ B and B − A together make A ∪ B.

Counting with Venn diagrams

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

n(A ∪ B ∪ C) = Σn(A) − Σn(A ∩ B) + n(A ∩ B ∩ C)

"Exactly one of A, B" = n(A) + n(B) − 2n(A ∩ B). "Neither A nor B" = n(U) − n(A ∪ B).

3. Relations

The Cartesian product A × B is the set of all ordered pairs (a, b) with a ∈ A and b ∈ B; n(A × B) = n(A)·n(B). A relation R from A to B is any subset of A × B. So the number of relations from A to B is 2^n(A)·n(B).

Types of relations on a set A

Type	Condition	Example on ℤ
Reflexive	(a, a) ∈ R for every a ∈ A	a − b is even
Symmetric	(a, b) ∈ R ⇒ (b, a) ∈ R	a + b is even
Transitive	(a, b), (b, c) ∈ R ⇒ (a, c) ∈ R	a < b
Anti-symmetric	(a, b), (b, a) ∈ R ⇒ a = b	a ≤ b
Equivalence	Reflexive + Symmetric + Transitive	a ≡ b (mod n)

Equivalence classes

An equivalence relation splits the set into disjoint equivalence classes (a partition). For example, a ≡ b (mod 5) on ℤ gives exactly 5 classes: [0], [1], [2], [3], [4]. Two classes are either identical or disjoint — never partly overlapping.

Common trap

The empty relation on a non-empty set is symmetric and transitive (there is nothing to break the rule) but not reflexive. The relation {(1,1),(2,2),(3,3)} on {1,2,3} is the smallest equivalence relation (the identity relation).

4. Functions

A function f : A → B is a relation in which every element of A has exactly one image in B. A is the domain, B the co-domain, and the set of actual images f(A) is the range (range ⊆ co-domain).

Types of functions

One-one (injective): different inputs give different outputs; f(x₁) = f(x₂) ⇒ x₁ = x₂. A strictly increasing or decreasing function is one-one.
Many-one: at least two inputs share an output (e.g. x², |x|, sin x on ℝ).
Onto (surjective): range = co-domain.
Bijective: one-one and onto. Only bijections have an inverse.

Left: each output has at most one arrow (one-one). Right: two inputs share an output (many-one), but every output is used (onto).

Counting functions (n(A) = n, n(B) = m)

Total functions A → B: mⁿ.
One-one functions: ^mP_n = m!/(m − n)! if m ≥ n, else 0.
Onto functions (n ≥ m): mⁿ − ^mC₁(m − 1)ⁿ + ^mC₂(m − 2)ⁿ − … (inclusion–exclusion).
Bijections (n = m): n!.

Domain rules you must remember

√g(x): need g(x) ≥ 0. 1/g(x): need g(x) ≠ 0.
log_a g(x): need g(x) > 0, a > 0, a ≠ 1.
sin⁻¹ g(x), cos⁻¹ g(x): need −1 ≤ g(x) ≤ 1.

Composition and inverse

(g∘f)(x) = g(f(x)) is defined when the range of f lies in the domain of g. Composition is associative but not commutative: usually g∘f ≠ f∘g. If f is a bijection, f⁻¹ exists and f∘f⁻¹ = f⁻¹∘f = identity. Also (g∘f)⁻¹ = f⁻¹∘g⁻¹.

Special functions

Even: f(−x) = f(x), graph symmetric about the y-axis. Odd: f(−x) = −f(x), symmetric about the origin.
Greatest integer function [x]: largest integer ≤ x. [2.7] = 2, [−2.7] = −3. Also x − 1 < [x] ≤ x and [x + n] = [x] + n for integer n.
Fractional part {x} = x − [x], with 0 ≤ {x} < 1; it is periodic with period 1.
Signum sgn x = 1, 0 or −1 according as x > 0, x = 0 or x < 0.

5. Formula Master Table

Quantity (n(A) = n, n(B) = m)	Formula
Subsets of A	2ⁿ
n(A × B)	n·m
Relations from A to B	2^nm
Relations on A	2^n²
Reflexive relations on A	2^{n² − n}
Symmetric relations on A	2^n(n+1)/2
Reflexive and symmetric relations	2^n(n−1)/2
Anti-symmetric relations	2ⁿ · 3^n(n−1)/2
Functions A → B	mⁿ
One-one functions (m ≥ n)	m!/(m − n)!
Onto functions A → B (n ≥ m)	Σ (−1)^r ^mC_r (m − r)ⁿ, r = 0 to m
Onto functions to a 2-element set	2ⁿ − 2
Bijections (n = m)	n!
Union of two sets	n(A) + n(B) − n(A∩B)

Domain and range of standard functions

Function	Domain	Range
√x	[0, ∞)	[0, ∞)
log x	(0, ∞)	ℝ
e^x	ℝ	(0, ∞)
\|x\|	ℝ	[0, ∞)
[x]	ℝ	ℤ
{x}	ℝ	[0, 1)
sin⁻¹x	[−1, 1]	[−π/2, π/2]
cos⁻¹x	[−1, 1]	[0, π]
tan⁻¹x	ℝ	(−π/2, π/2)
1/x	ℝ − {0}	ℝ − {0}

6. Mind Map

Sets, Relations & Functions

Sets

Types: empty, finite, power set
∪, ∩, complement, difference
De Morgan's laws
Venn counting

Relations

Subset of A × B
Reflexive, symmetric, transitive
Equivalence → partition
Counting formulas

Functions

Domain, co-domain, range
One-one, onto, bijective
Counting mⁿ, ^mP_n

Operations

Composition g∘f
Inverse (bijection only)
Even / odd / periodic
[x], {x}, sgn x

7. Tips and Tricks

Shortcuts that save time

Check one-one fast: if f′(x) keeps one sign (> 0 or < 0) on the whole domain, f is one-one. Polynomials of even degree are never one-one on ℝ.
Check onto fast: find the range (use the graph, or solve y = f(x) for x and see which y are allowed) and compare it with the co-domain.
Functions of the form f(x) = (ax + b)/(cx − a) are their own inverse, so f(f(x)) = x.
For "is R an equivalence relation?" questions, try to find a counter-example with small numbers or with fractions like ½ — one counter-example is enough to rule out a property.
For domain questions with log inside √, solve the inner log inequality carefully: log_a t ≥ 0 means t ≥ 1 if a > 1, but 0 < t ≤ 1 if 0 < a < 1.

Mistakes students make

Writing [−2.7] = −2. The correct value is −3 (the greatest integer less than or equal to −2.7).
Assuming g∘f = f∘g. Always compute both separately.
Forgetting that the range must equal the given co-domain for "onto" — f(x) = x² is onto from ℝ to [0, ∞) but not onto from ℝ to ℝ.

8. Important One-liners

The empty set is a subset of every set, and every set is a subset of itself.
A set with n elements has 2ⁿ subsets; its power set has 2ⁿ elements.
If A ⊆ B, then A ∪ B = B and A ∩ B = A.
Every equivalence relation on a set produces a partition of that set, and every partition produces an equivalence relation.
The identity relation is reflexive, symmetric, transitive and anti-symmetric.
A function has an inverse if and only if it is a bijection.
If f and g are both one-one, g∘f is one-one; if both are onto, g∘f is onto.
The only function that is both even and odd is f(x) = 0.
Any function defined on a symmetric domain can be written as the sum of an even and an odd function.
x − [x] = {x} always lies in [0, 1).
The graph of f⁻¹ is the mirror image of the graph of f in the line y = x.

9. PYQ-pattern Questions with Solutions

These questions are original and modelled on concepts that are frequently tested in JEE Main.

Q1Medium

The number of symmetric relations on the set A = {1, 2, 3} is:

Show solution

A × A has 9 ordered pairs: 3 diagonal pairs (a, a) and 6 off-diagonal pairs forming 3 "mirror" couples {(a, b), (b, a)}. In a symmetric relation, each diagonal pair can be included or not (2 choices each), and each mirror couple must be taken together or left together (2 choices each).

Number = 2³ × 2³ = 2⁶ = 64, which matches 2^n(n+1)/2 with n = 3.

Answer: (B) 64

Q2Medium

The domain of f(x) = √(log_1/2(x − 1)) is:

(1, 2]
[2, ∞)
(1, ∞)
[1, 2]

Show solution

We need log_1/2(x − 1) ≥ 0 and x − 1 > 0. Since the base ½ is less than 1, log_1/2 t ≥ 0 means 0 < t ≤ 1.

So 0 < x − 1 ≤ 1, that is 1 < x ≤ 2.

Answer: (A) (1, 2]

Trap: students who treat the base as greater than 1 get x ≥ 2, which is option (B).

Q3Hard

The number of onto functions from A = {1, 2, 3, 4} to B = {a, b, c} is:

Show solution

Total functions = 3⁴ = 81. Remove those that miss at least one element of B using inclusion–exclusion:

81 − ³C₁·2⁴ + ³C₂·1⁴ = 81 − 48 + 3 = 36

Check another way: choose which two elements of A share an image (⁴C₂ = 6 ways), then map the three "groups" onto a, b, c in 3! = 6 ways: 6 × 6 = 36.

Answer: (B) 36

Q4Hard

Let R be the relation on ℝ defined by R = {(a, b) : a ≤ b²}. Then R is:

Reflexive only
Symmetric only
Transitive only
Neither reflexive, nor symmetric, nor transitive

Show solution

Reflexive? Take a = ½: is ½ ≤ ¼? No. So R is not reflexive.

Symmetric? (1, 2) ∈ R since 1 ≤ 4, but (2, 1) ∉ R since 2 ≤ 1 is false.

Transitive? (3, 2) ∈ R (3 ≤ 4) and (2, 1.5) ∈ R (2 ≤ 2.25), but (3, 1.5) ∉ R because 3 ≤ 2.25 is false.

Answer: (D)

Q5Numerical

If f(x) = (x + 4)/(x − 1), x ≠ 1, find the value of f(f(f(2))).

Show solution

f is of the form (ax + b)/(cx − a) with a = 1, b = 4, c = 1, so f is its own inverse: f(f(x)) = x. Check: f(f(x)) = [(x+4)/(x−1) + 4] / [(x+4)/(x−1) − 1] = (5x)/(5) = x.

Hence f(f(f(2))) = f(2) = 6/1 = 6.

Answer: 6

Q6Medium

The function f : ℝ → ℝ, f(x) = x³ + x is:

One-one but not onto
Onto but not one-one
Both one-one and onto
Neither one-one nor onto

Show solution

f′(x) = 3x² + 1 > 0 for all x, so f is strictly increasing and therefore one-one.

As x → ∞, f(x) → ∞ and as x → −∞, f(x) → −∞. Being continuous, f takes every real value, so the range is ℝ = co-domain: f is onto.

Answer: (C)

Q7Numerical

In a class of 60 students, 35 like Mathematics, 30 like Physics and 12 like neither subject. How many students like both subjects?

Show solution

Students who like at least one subject = 60 − 12 = 48 = n(M ∪ P).

n(M ∩ P) = n(M) + n(P) − n(M ∪ P) = 35 + 30 − 48 = 17.

Answer: 17

Q8Easy

Let R be the relation on ℤ given by a R b if and only if a − b is divisible by 5. The number of distinct equivalence classes of R is:

1
4
5
Infinite

Show solution

R is congruence modulo 5, which is an equivalence relation. Every integer leaves one of the remainders 0, 1, 2, 3 or 4 on division by 5, and integers with the same remainder form one class.

Answer: (C) 5

10. Quick Revision Summary

Remember before the exam

n(P(A)) = 2ⁿ; relations on A = 2^n²; reflexive = 2^n²−n; symmetric = 2^n(n+1)/2.
Equivalence = reflexive + symmetric + transitive; its classes form a partition.
Functions A → B: mⁿ; one-one: ^mP_n; onto: inclusion–exclusion; bijections: n!.
Domain: √ needs ≥ 0, log needs > 0, sin⁻¹/cos⁻¹ need values in [−1, 1].
Inverse exists only for bijections; (g∘f)⁻¹ = f⁻¹∘g⁻¹.
[−2.7] = −3, {x} ∈ [0, 1), and (ax + b)/(cx − a) is self-inverse.

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