JEE Mains 2026 Revision Capsule – Relations & Functions (Mathematics)

A focused revision of Relations and Functions with all key definitions, properties, classifications, and JEE-shortcuts presented in a crisp, exam-friendly form.

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1. Cartesian Product

A × B = {(a, b) | a ∈ A, b ∈ B}
If n(A)=m and n(B)=n → n(A × B) = m·n

Ordered pairs matter: (a, b) ≠ (b, a)

2. Relation

A relation R from A to B is any subset of A × B.

Types of Relations

Reflexive: (a, a) ∈ R for all a ∈ A
Symmetric: (a, b) ∈ R ⇒ (b, a) ∈ R
Transitive: (a, b), (b, c) ∈ R ⇒ (a, c) ∈ R
Equivalence Relation: Reflexive + Symmetric + Transitive

Equivalence relations partition a set into disjoint equivalence classes.

3. Function (Mapping)

A function f: A → B is a relation where: • each a ∈ A has exactly one image in B • no element of A maps to two different elements

Domain: all elements of A
Codomain: B
Range: actual outputs under f

4. Types of Functions

One-One (Injective): different inputs → different outputs
Onto (Surjective): Range = Codomain
Bijection: both one-one and onto

Bijections have inverses.

Special Types

• Constant function: f(x)=c
• Identity function: f(x)=x
• Even function: f(−x)=f(x)
• Odd function: f(−x)=−f(x)

5. Composition of Functions

(f ∘ g)(x) = f(g(x))
Composition is associative, not necessarily commutative.

Domain of (f∘g): elements x where g(x) is defined and f(g(x)) is defined.

6. Inverse of a Function

f⁻¹ exists only if f is bijective.
f(f⁻¹(x)) = x and f⁻¹(f(x)) = x

7. JEE High-Yield Results

• For a finite set A: number of relations on A = 2ⁿ² where n = |A|
• Number of functions: |B|^|A|
• Bijective functions only possible when |A| = |B|
• Equivalence relations ↔ partitions of a set

8. JEE Problem Patterns

✔ Identify relation type (R/S/T)
✔ Check if a function is injective/surjective
✔ Find range/domain of given algebraic functions
✔ Find number of relations or functions for given sets
✔ Solve composite function or inverse questions

9. Quick Self-Test

1. Check whether R = {(1,1),(2,2),(1,2),(2,1)} is an equivalence relation.
2. If f(x)=3x−5, find f⁻¹(x).
3. How many functions from A={1,2,3} to B={a,b}?
4. State whether f(x)=x² is injective on R.
5. If f(x)=|x|, is it even/odd?

Relations and Functions act as the blueprint of algebra. Mastering them strengthens all later topics: trigonometry, calculus, and coordinate geometry.