Unit 1 • Mathematics

Sets, Relations & Functions

Foundation of Modern Mathematics | JEE Main 2025

Very High Weightage • 3–4 Questions Expected
Sets
Operations
Relations
Functions
JEE Tips

Sets

A set is a well-defined collection of distinct objects. Objects are called elements or members.
Representation

Roster form: {1, 2, 3, 4}
Set-builder form: {x : x ∈ N, x ≤ 5}

Types of Sets
  • Empty set (∅)
  • Singleton set
  • Finite / Infinite
  • Equal sets (same elements)
  • Equivalent sets (same cardinality)

Power Set

Set of all subsets of A → P(A)
If A has n elements → |P(A)| = 2ⁿ

Universal Set (U)

Set containing all objects under consideration.

Venn Diagrams

Graphical representation of sets using circles.

Visual Example: Venn Diagram

Imagine two overlapping circles representing sets A and B:

  • The overlapping region represents A ∩ B
  • The total area covered by both circles represents A ∪ B
  • The area in A but not in B represents A - B
Practice Question

If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find:

  1. A ∪ B
  2. A ∩ B
  3. A - B

1. A ∪ B = {1, 2, 3, 4, 5, 6}

2. A ∩ B = {3, 4}

3. A - B = {1, 2}

Operations on Sets

Union (A ∪ B)

x ∈ A ∪ B ⇔ x ∈ A or x ∈ B

A ∪ B = {x : x ∈ A ∨ x ∈ B}
Intersection (A ∩ B)

x ∈ A ∩ B ⇔ x ∈ A and x ∈ B

A ∩ B = {x : x ∈ A ∧ x ∈ B}
Difference (A – B)

A – B = {x : x ∈ A, x ∉ B}

Complement (A')

A' = U – A = {x ∈ U : x ∉ A}

Important Identities

  • A ∪ A = A, A ∩ A = A (Idempotent)
  • A ∪ ∅ = A, A ∩ U = A
  • De Morgan’s Laws:
    (A ∪ B)' = A' ∩ B'
    (A ∩ B)' = A' ∪ B'
  • |A ∪ B| = |A| + |B| – |A ∩ B|

Quick Reference

Union: All elements from both sets
Intersection: Common elements only
Difference: Elements in first but not in second
Complement: Elements not in the set

Relations

A relation R from set A to set B is a subset of A × B.
If (a, b) ∈ R → a is related to b (a R b)
Types of Relations
RelationCondition
Reflexive(a,a) ∈ R ∀ a ∈ A
SymmetricIf (a,b) ∈ R ⇒ (b,a) ∈ R
TransitiveIf (a,b), (b,c) ∈ R ⇒ (a,c) ∈ R
EquivalenceReflexive + Symmetric + Transitive
Equivalence Relation Example

“Congruence modulo m” on integers
“is parallel to” on lines
“is similar to” on triangles

Visual Example: Relation Types

Think of relations as connections between points:

  • Reflexive: Every point has a loop to itself
  • Symmetric: If there's an arrow from A to B, there's also one from B to A
  • Transitive: If A→B and B→C, then there must be A→C

Functions

A function f : A → B is a special relation where every element in domain A is related to exactly one element in codomain B.
Types of Functions
  • One-One (Injective): f(a) = f(b) ⇒ a = b
  • Many-One: Different elements map to same
  • Onto (Surjective): Every element in B is image
  • Into: Not onto
  • Bijective: Both one-one and onto
Common Functions

Identity: f(x) = x
Constant: f(x) = c
Polynomial, Rational
Modulus, Signum, Greatest Integer, Fractional Part

Key Formulas

• Number of injective functions from A to B: ⁿPₘ = n!/(n-m)!
• Number of functions: mⁿ
• Number of bijections (if |A| = |B| = n): n!

Composition of Functions

(f ∘ g)(x) = f(g(x))
Note: f ∘ g ≠ g ∘ f in general
Both are associative

Inverse Function

Exists only if f is bijective
f⁻¹ ∘ f = I, f ∘ f⁻¹ = I

Practice Question

Determine if the function f: R → R defined by f(x) = x² is:

  1. One-one
  2. Onto

1. Not one-one because f(2) = f(-2) = 4 but 2 ≠ -2

2. Not onto because negative numbers are not in the range

JEE Main Weightage & Tips

One of the most important and scoring chapters. Usually 3–4 questions appear every year.

WeightageVery High (10–12 marks)

Must Remember for JEE

  • All properties of equivalence relations
  • Injective/Surjective tests (horizontal line, algebraic)
  • Number of relations/functions formulas
  • Inverse trigonometric functions domain/range (comes later but linked)
  • Graph of modulus, signum, greatest integer functions

Quick Reference

Power Set: |P(A)| = 2ⁿ
Union Formula: |A∪B| = |A| + |B| - |A∩B|
Functions: mⁿ total functions
Injective: ⁿPₘ = n!/(n-m)!
Exam Strategy
  • Focus on definitions and properties - these are frequently tested
  • Practice problems on finding number of relations/functions
  • Master the difference between various types of relations
  • Understand function composition and inverse functions thoroughly