JEE 2026 – Mathematics
SETS, RELATIONS & FUNCTIONS
(A–Z BENCHMARK NOTES)
CHAPTER OVERVIEW
This chapter forms the foundation of entire higher mathematics.
Every advanced topic — calculus, algebra, probability, matrices — is built on:
- Set language
- Relations as logical connections
- Functions as mathematical machines
PART A — SETS
1. What is a Set?
A set is a well-defined collection of distinct objects, called elements.
“Well-defined” means: No ambiguity about whether an object belongs to the set or not.
“Well-defined” means: No ambiguity about whether an object belongs to the set or not.
✔ Set of even natural numbers less than 10
✖ Set of intelligent students (not well-defined)
2. Representation of Sets
| Method | Description | Example |
|---|---|---|
| Roster Form | Elements listed explicitly | $\{1,2,3,4\}$ |
| Set Builder Form | Property-based description | $\{x \mid x \in \mathbb{N}, x \le 4\}$ |
3. Types of Sets (Complete Classification)
| Type | Definition |
|---|---|
| Empty Set | No elements ($\emptyset$) |
| Singleton Set | Exactly one element |
| Finite Set | Limited number of elements |
| Infinite Set | Unlimited elements |
| Equal Sets | Same elements |
| Equivalent Sets | Same number of elements |
4. Subsets
If every element of $A$ belongs to $B$, then $A \subseteq B$.
Number of subsets of a set with $n$ elements = $2^n$
If $A=\{a,b,c\}$, total subsets = $2^3=8$
5. Proper & Improper Subsets
| Type | Meaning |
|---|---|
| Proper Subset | $A \subset B$ and $A \ne B$ |
| Improper Subset | Set itself |
6. Power Set
The power set of $A$, denoted $P(A)$, is the set of all subsets of $A$.
If $A=\{1,2\}$,
$P(A)=\{\emptyset,\{1\},\{2\},\{1,2\}\}$
7. Universal Set
The universal set ($U$) contains all objects under consideration.
8. Operations on Sets
| Operation | Symbol | Meaning |
|---|---|---|
| Union | $A \cup B$ | Elements in $A$ or $B$ |
| Intersection | $A \cap B$ | Common elements |
| Difference | $A-B$ | In $A$ but not in $B$ |
| Complement | $A'$ | Not in $A$ |
9. Laws of Sets
De Morgan’s Laws
$(A \cup B)' = A' \cap B'$
$(A \cap B)' = A' \cup B'$
$(A \cup B)' = A' \cap B'$
$(A \cap B)' = A' \cup B'$
⚠ JEE Trap:
De Morgan’s laws are often tested in hidden form inside word problems.
PART B — RELATIONS
10. Ordered Pair
$(a,b)$ is an ordered pair where order matters:
$(a,b) \ne (b,a)$ unless $a=b$
11. Cartesian Product
$A \times B = \{(a,b) \mid a \in A,\ b \in B\}$
If $A=\{1,2\}$ and $B=\{x,y\}$,
$A\times B=\{(1,x),(1,y),(2,x),(2,y)\}$
12. Relation
A relation from $A$ to $B$ is any subset of $A \times B$.
13. Types of Relations
| Relation | Condition |
|---|---|
| Reflexive | $(a,a) \in R$ |
| Symmetric | $(a,b) \Rightarrow (b,a)$ |
| Transitive | $(a,b),(b,c) \Rightarrow (a,c)$ |
14. Equivalence Relation
A relation is an equivalence relation iff it is:
Reflexive + Symmetric + Transitive
Reflexive + Symmetric + Transitive
$a \equiv b \pmod{3}$ is an equivalence relation
15. Equivalence Class
The equivalence class of $a$ is:
$[a]=\{x \mid xRa\}$
Equivalence relations partition a set into disjoint equivalence classes.
PART C — FUNCTIONS
16. Function (Most Important Definition)
A function $f:A \to B$ assigns exactly one output in $B$ to each input in $A$.
17. Domain, Codomain, Range
| Term | Meaning |
|---|---|
| Domain | All possible inputs |
| Codomain | All allowed outputs |
| Range | Actual outputs |
18. Types of Functions
| Type | Key Property |
|---|---|
| One-One | $f(x_1)=f(x_2)\Rightarrow x_1=x_2$ |
| Many-One | Different inputs → same output |
| Onto | Range = Codomain |
| Into | Range ⊂ Codomain |
| Bijective | One-One + Onto |
| Constant | Same output for all inputs |
| Identity | $f(x)=x$ |
19. Algebra of Functions
$(f+g)(x)=f(x)+g(x)$
$(f-g)(x)=f(x)-g(x)$
$(fg)(x)=f(x)g(x)$
$\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}$
$(f-g)(x)=f(x)-g(x)$
$(fg)(x)=f(x)g(x)$
$\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}$
20. Composition of Functions
$(f\circ g)(x)=f(g(x))$
⚠ Order matters:
$f\circ g \ne g\circ f$ (in general)
21. Invertible Functions
A function has an inverse if and only if it is bijective.
$f(x)=2x+3$
$f^{-1}(x)=\dfrac{x-3}{2}$
22. Domain & Range (Critical for JEE)
For domain:
- Denominator $\ne 0$
- Even root $\ge 0$
- Log argument $>0$
FINAL JEE BENCHMARK SUMMARY
If you can:
- Translate word problems into set language
- Identify equivalence relations
- Find domain & range confidently
- Decide one-one / onto without guessing
- Find inverse functions step-by-step