Sets, Relations and Functions – JEE Mains Mathematics

• by Zenith LMS
• Updated April 2025

1. Introduction to Sets

A set is a well-defined collection of distinct objects. Sets are usually represented by capital letters.

• Roster form: Listing all elements, e.g., A = {1, 2, 3}
• Set-builder form: Describes properties, e.g., A = {x | x is a natural number < 5}

2. Types of Sets

• Empty set
• Finite and Infinite sets
• Equal sets
• Subsets and Power Set
• Universal set and Complement of a set

3. Set Operations

• Union (A ∪ B): Elements in A or B or both
• Intersection (A ∩ B): Elements common to both A and B
• Complement (Aʹ): Elements not in A
• Algebra of sets: Includes laws like associative, commutative, distributive, De Morgan’s laws

4. Cartesian Product and Relations

• Cartesian product (A × B): Set of all ordered pairs (a, b) where a ∈ A and b ∈ B
• Relation: A subset of Cartesian product A × B
• Types of relations: Reflexive, Symmetric, Transitive, and Equivalence

5. Functions

• Function: A relation where every element of the domain has exactly one image in the co-domain
• Types: One-one (injective), Onto (surjective), Into, Bijective (both one-one and onto)
• Composition of functions: If f: A → B and g: B → C, then g ∘ f: A → C