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Sets, Relations and Functions Formulas For JEE 2026
Sets, Relations, and Functions is one of the most important chapters in the Mathematics portion of the JEE exam, as it acts as a foundation for other important topics like algebra, calculus, and probability. Students can expect around 1 to 2 questions from this chapter in the JEE Main exam, making it an important one to master.
This chapter deals with important concepts like set operations, Venn diagrams, different types of relations, different types of functions, and composition of functions. It also deals with important formulas and examples, which help students to understand concepts in a better manner.
Sets Formulas for JEE
A set is a well-defined collection of distinct objects. These objects are called elements or members of the set. "Well-defined" means there is no ambiguity about whether something belongs to the set or not.
For example, "the set of all vowels in English" is well-defined (a, e, i, o, u), but "the set of all tall people" is not (what counts as tall?).
Set: A well-defined collection of distinct objects, usually denoted by capital letters like $$A$$, $$B$$, $$C$$.
Element: An object belonging to a set. If $$x$$ belongs to set $$A$$, we write $$x \in A$$. If not, we write $$x \notin A$$.
Ways to Represent a Set
Set Notation
- Roster (Listing) method: List all elements inside braces.
$$A = \{1, 2, 3, 4, 5\}$$, $$B = \{a, e, i, o, u\}$$ - Set-builder method: Describe the property that elements must satisfy.
$$A = \{x : x \text{ is a natural number}, x \leq 5\}$$
(Read: "$$A$$ is the set of all $$x$$ such that $$x$$ is a natural number and $$x \leq 5$$")
Standard Number Sets
Important Number Sets
| Symbol | Name | Contains |
|---|---|---|
| $$\mathbb{N}$$ | Natural numbers | $$\{1, 2, 3, 4, \ldots\}$$ |
| $$\mathbb{W}$$ | Whole numbers | $$\{0, 1, 2, 3, \ldots\}$$ |
| $$\mathbb{Z}$$ | Integers | $$\{\ldots, -2, -1, 0, 1, 2, \ldots\}$$ |
| $$\mathbb{Q}$$ | Rational numbers | all fractions $$\frac{p}{q}$$ where $$p, q \in \mathbb{Z}$$, $$q \neq 0$$ |
| $$\mathbb{R}$$ | Real numbers | all numbers on the number line |
These sets are nested: $$\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$$. Every natural number is also a whole number, every whole number is an integer, and so on.
Types of Sets Formulas
Special Types of Sets
- Empty set ($$\emptyset$$ or $$\{\}$$): A set with no elements. E.g., $$\{x : x^2 = -1, x \in \mathbb{R}\} = \emptyset$$.
- Singleton set: A set with exactly one element. E.g., $$\{5\}$$.
- Finite set: Has a countable number of elements. E.g., $$\{2, 4, 6, 8\}$$.
- Infinite set: Has unlimited elements. E.g., $$\mathbb{N} = \{1, 2, 3, \ldots\}$$.
- Equal sets: $$A = B$$ if they have exactly the same elements.
- Universal set ($$U$$): The set containing all elements under consideration.
Subsets Formulas for JEE
If every element of set $$A$$ is also in set $$B$$, then $$A$$ is a subset of $$B$$. Think of it as "$$A$$ fits entirely inside $$B$$."
Subset Rules
- $$A \subseteq B$$ means every element of $$A$$ is in $$B$$ ($$A$$ is a subset of $$B$$).
- $$A \subset B$$ means $$A \subseteq B$$ and $$A \neq B$$ ($$A$$ is a proper subset).
- The empty set $$\emptyset$$ is a subset of every set.
- Every set is a subset of itself: $$A \subseteq A$$.
- If $$A$$ has $$n$$ elements, the total number of subsets is $$2^n$$.
Worked Example
Find all subsets of $$A = \{1, 2\}$$.
$$n = 2$$, so number of subsets $$= 2^2 = 4$$.
Subsets: $$\emptyset, \{1\}, \{2\}, \{1, 2\}$$
Proper subsets (excluding $$A$$ itself): $$\emptyset, \{1\}, \{2\}$$ — there are $$2^n - 1 = 3$$.
Set Operations Formulas for JEE
Just like we have operations on numbers (addition, subtraction), we have operations on sets that combine or compare them.
Union ($$A \cup B$$)
The set of all elements that are in $$A$$ or $$B$$ (or both).
$$$A \cup B = \{x : x \in A \text{ or } x \in B\}$$$
Intersection ($$A \cap B$$)
The set of all elements that are in both $$A$$ and $$B$$.
$$$A \cap B = \{x : x \in A \text{ and } x \in B\}$$$
Difference ($$A - B$$)
The set of elements in $$A$$ but not in $$B$$.
$$$A - B = \{x : x \in A \text{ and } x \notin B\}$$$
Complement ($$A'$$ or $$A^c$$)
The set of all elements in the universal set $$U$$ that are not in $$A$$.
$$$A' = U - A = \{x : x \in U \text{ and } x \notin A\}$$$
Worked Example
Let $$A = \{1, 2, 3, 4\}$$, $$B = \{3, 4, 5, 6\}$$, $$U = \{1, 2, 3, 4, 5, 6, 7\}$$.
$$A \cup B = \{1, 2, 3, 4, 5, 6\}$$
$$A \cap B = \{3, 4\}$$
$$A - B = \{1, 2\}$$
$$A' = \{5, 6, 7\}$$
Cardinality Formulas for Sets
Cardinality (Inclusion-Exclusion) Formulas
For finite sets, $$n(A)$$ denotes the number of elements in $$A$$.
For two sets:
$$$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$$
For three sets:
$$$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$$$
Worked Example
In a class of 60 students, 35 play cricket, 25 play football, and 10 play both. How many play neither?
$$n(C \cup F) = n(C) + n(F) - n(C \cap F) = 35 + 25 - 10 = 50$$
Students playing neither $$= 60 - 50 =$$ 10
De Morgan's Laws
- $$(A \cup B)' = A' \cap B'$$ (complement of union = intersection of complements)
- $$(A \cap B)' = A' \cup B'$$ (complement of intersection = union of complements)
Tip: De Morgan's Laws are frequently tested in JEE. A quick way to remember: when you take the complement, $$\cup$$ becomes $$\cap$$ and vice versa — the operations "flip."
Relations Formulas for JEE
A relation describes a connection between elements of two sets. Formally, a relation from set $$A$$ to set $$B$$ is a subset of the Cartesian product $$A \times B$$.
Cartesian Product: $$A \times B = \{(a, b) : a \in A, b \in B\}$$ — the set of all ordered pairs where the first element comes from $$A$$ and the second from $$B$$.
Worked Example
If $$A = \{1, 2\}$$ and $$B = \{a, b\}$$, find $$A \times B$$.
$$A \times B = \{(1, a), (1, b), (2, a), (2, b)\}$$
Note: $$n(A \times B) = n(A) \times n(B) = 2 \times 2 = 4$$
Relation: A relation $$R$$ from $$A$$ to $$B$$ is any subset of $$A \times B$$. If $$(a, b) \in R$$, we write $$a \, R \, b$$ (read: "$$a$$ is related to $$b$$").
Types of Relations
When a relation is from a set $$A$$ to itself ($$R \subseteq A \times A$$), it can have special properties:
Properties of Relations
- Reflexive: Every element is related to itself. $$\forall a \in A$$: $$(a, a) \in R$$.
E.g., "is equal to" — every number equals itself. - Symmetric: If $$a$$ is related to $$b$$, then $$b$$ is related to $$a$$. $$(a, b) \in R \Rightarrow (b, a) \in R$$.
E.g., "is a sibling of" — if A is B's sibling, B is A's sibling. - Transitive: If $$a$$ is related to $$b$$ and $$b$$ is related to $$c$$, then $$a$$ is related to $$c$$. $$(a, b) \in R$$ and $$(b, c) \in R \Rightarrow (a, c) \in R$$.
E.g., "is less than" — if $$a < b$$ and $$b < c$$, then $$a < c$$. - Equivalence relation: A relation that is reflexive, symmetric, and transitive.
E.g., "is equal to", "is congruent to (mod $$n$$)".
Worked Example
On $$A = \{1, 2, 3\}$$, let $$R = \{(1,1), (2,2), (3,3), (1,2), (2,1)\}$$. Check its properties.
Reflexive? Yes — $$(1,1), (2,2), (3,3)$$ are all present.
Symmetric? Yes — $$(1,2) \in R$$ and $$(2,1) \in R$$. No other asymmetric pairs.
Transitive? Check: $$(1,2)$$ and $$(2,1) \in R \Rightarrow (1,1)$$ must be in $$R$$. Yes. Also $$(2,1)$$ and $$(1,2) \Rightarrow (2,2)$$ must be in $$R$$. Yes.
$$R$$ is an equivalence relation.
Functions Formulas for JEE
A function is a special type of relation where every element in the first set (called the domain) is related to exactly one element in the second set (called the codomain). Think of it as a rule that assigns each input exactly one output.
Function: A relation $$f: A \to B$$ such that every element of $$A$$ has exactly one image in $$B$$. We write $$f(x) = y$$, meaning input $$x$$ gives output $$y$$.
Domain: The set of all possible inputs ($$A$$).
Codomain: The set in which outputs are allowed to lie ($$B$$).
Range: The set of actual outputs: $$\{f(x) : x \in A\} \subseteq B$$.
Note: Range $$\subseteq$$ Codomain, always. They are equal only when the function is "onto" (surjective).
Types of Functions
Important Function Types
- One-one (Injective): Different inputs give different outputs.
$$f(a) = f(b) \Rightarrow a = b$$
E.g., $$f(x) = 2x$$ is one-one. $$f(x) = x^2$$ is not one-one (since $$f(2) = f(-2) = 4$$). - Onto (Surjective): Every element of the codomain has at least one pre-image.
Range $$=$$ Codomain
E.g., $$f: \mathbb{R} \to \mathbb{R}$$, $$f(x) = 2x$$ is onto. $$f(x) = x^2$$ from $$\mathbb{R} \to \mathbb{R}$$ is not onto (negative numbers have no pre-image). - Bijective: Both one-one and onto. A bijection has an inverse function.
Number of Functions Formulas
Counting Functions
If $$n(A) = m$$ and $$n(B) = n$$:
- Total functions from $$A$$ to $$B$$: $$n^m$$
- One-one functions (requires $$m \leq n$$): $$\dfrac{n!}{(n-m)!}$$
- Onto functions (requires $$m \geq n$$): found using inclusion-exclusion
- Bijections (requires $$m = n$$): $$n!$$
Worked Example
If $$A = \{1, 2, 3\}$$ and $$B = \{a, b\}$$, how many functions exist from $$A$$ to $$B$$?
Total functions $$= n^m = 2^3 =$$ 8
One-one functions: Not possible since $$m = 3 > n = 2$$ (pigeonhole principle — 3 elements can't all map to distinct values in a 2-element set).
Composition and Inverse Function Formulas
Composition: If $$f: A \to B$$ and $$g: B \to C$$, then $$g \circ f: A \to C$$ is defined as $$(g \circ f)(x) = g(f(x))$$.
Properties of Composition
- $$g \circ f \neq f \circ g$$ in general (not commutative)
- $$(h \circ g) \circ f = h \circ (g \circ f)$$ (associative)
- If $$f$$ and $$g$$ are one-one, then $$g \circ f$$ is one-one
- If $$f$$ and $$g$$ are onto, then $$g \circ f$$ is onto
Inverse Function: If $$f: A \to B$$ is a bijection, then $$f^{-1}: B \to A$$ exists such that $$f^{-1}(f(x)) = x$$ and $$f(f^{-1}(y)) = y$$.
Worked Example
Let $$f(x) = 2x + 3$$. Find $$f^{-1}(x)$$.
Let $$y = 2x + 3$$.
Solve for $$x$$: $$x = \frac{y - 3}{2}$$.
So $$f^{-1}(x) = \frac{x - 3}{2}$$.
Verify: $$f(f^{-1}(x)) = f\!\left(\frac{x-3}{2}\right) = 2 \cdot \frac{x-3}{2} + 3 = x$$ ✓
Tip: To find $$f^{-1}$$: (1) write $$y = f(x)$$, (2) solve for $$x$$ in terms of $$y$$, (3) replace $$y$$ with $$x$$. This only works when $$f$$ is bijective. In JEE, always verify by checking $$f(f^{-1}(x)) = x$$.
Sets, Relations and Functions Formulas For JEE 2026: Conclusion
Sets, relations, and functions formulas for JEE 2026 are very important for building a strong base in math. This chapter talks about important ideas like set operations, relations, types of functions, and how to put functions together. Students can answer questions more easily and do better on the test if they understand these ideas clearly and go over the formulas often.
In the last stage of getting ready, you should practice important formulas, review important properties, and answer questions that are based on concepts. Instead of going over everything again, use short notes and formula sheets to quickly review. This chapter can be one of the easiest parts of JEE Maths 2026 to score well on if you practice a lot and understand it well.
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