Definition & Representation of Sets
A set is a well-defined collection of distinct objects. "Well-defined" means that given any object, it should be clear whether that object belongs to the collection or not. The objects in a set are called its elements or members.
1.1 Representation of Sets
Most Repeated PYQ — Set Forms
There are two main ways to represent a set:
- Roster (Tabular) Form: All elements are listed, separated by commas, enclosed in braces. E.g., $A = \{1, 2, 3, 4\}$. Elements are never repeated, and order doesn't matter.
- Set-Builder Form: All elements possess a single common property. E.g., $A = \{x : x \text{ is a natural number less than } 5\}$.
PYQ — JEE Main 2021 (Aug 26 Shift 1)
Question: The set builder form of the set $A = \{1, 3, 5, 7, 9\}$ is:
(a) $\{x : x \text{ is an odd natural number less than } 10\}$
(b) $\{x : x \text{ is an integer}\}$
(c) $\{x : x \text{ is a prime number}\}$
(d) $\{x : x \text{ is an odd number}\}$
Solution:
The elements are $1, 3, 5, 7, 9$.
All are odd natural numbers. The largest is 9, so they are all odd natural numbers less than 10.
Answer: (a)
Types of Sets
Sets can be classified based on the number of elements they contain.
2.1 Empty, Singleton, and Finite/Infinite Sets
Important Definitions
- Empty (Null) Set ($\emptyset$): A set containing no elements. E.g., $A = \{x : x^2 = 4 \text{ and } x \text{ is odd}\}$.
- Singleton Set: A set with exactly one element. E.g., $A = \{5\}$.
- Finite Set: A set with a finite number of elements.
- Infinite Set: A set with an infinite number of elements. E.g., the set of natural numbers $N = \{1, 2, 3, ...\}$.
2.2 Equal and Equivalent Sets
- Equal Sets: Two sets $A$ and $B$ are equal ($A = B$) if they have exactly the same elements.
- Equivalent Sets: Two finite sets are equivalent if they have the same number of elements (i.e., $n(A) = n(B)$).
PYQ — JEE Main 2019 (Apr 8 Shift 2)
Question: Which of the following is an empty set?
(a) $\{x : x \in \mathbb{N} \text{ and } x^2 = 4\}$
(b) $\{x : x \in \mathbb{N} \text{ and } x^2 = 9\}$
(c) $\{x : x \in \mathbb{N} \text{ and } x^2 = 2\}$
(d) $\{x : x \in \mathbb{N} \text{ and } x \text{ is even}\}$
Solution:
For $x^2 = 4$, $x = 2 \in \mathbb{N}$. Set = $\{2\}$.
For $x^2 = 9$, $x = 3 \in \mathbb{N}$. Set = $\{3\}$.
For $x^2 = 2$, $x = \pm\sqrt{2}$. Neither is a natural number. This is an empty set.
Answer: (c)
Subsets & Superset
Set $A$ is a subset of set $B$ ($A \subseteq B$) if every element of $A$ is also an element of $B$. $B$ is then called the superset of $A$.
3.1 Subset Properties
Most Repeated PYQ — Subset Count
- The empty set is a subset of every set.
- Every set is a subset of itself.
- If a set has $n$ elements, the total number of subsets is $2^n$.
- The total number of proper subsets is $2^n - 1$.
3.2 Intervals as Subsets of Real Numbers
For $a, b \in \mathbb{R}$ and $a < b$:
- Closed Interval: $[a, b] = \{x : a \le x \le b\}$ (includes endpoints)
- Open Interval: $(a, b) = \{x : a < x < b\}$ (excludes endpoints)
- Semi-open/ Semi-closed: $[a, b) = \{x : a \le x < b\}$
PYQ — JEE Main 2020 (Sep 4 Shift 1)
Question: The number of proper subsets of a set with 5 elements is:
(a) 32
(b) 31
(c) 25
(d) 10
Solution:
Given $n = 5$.
Total subsets = $2^n = 2^5 = 32$.
Proper subsets = Total subsets - 1 (excluding the set itself)
Answer: (b) 31
Power Set
The set of all subsets of a set $A$ is called the power set of $A$, denoted by $P(A)$.
4.1 Properties of Power Set
Most Repeated PYQ — Power Set Cardinality
- If $n(A) = m$, then $n(P(A)) = 2^m$.
- The power set of an empty set has exactly 1 element: $P(\emptyset) = \{\emptyset\}$. Thus $n(P(\emptyset)) = 1 = 2^0$.
- Power set is always a set of sets. E.g., If $A = \{1, 2\}$, then $P(A) = \{\emptyset, \{1\}, \{2\}, \{1, 2\}\}$.
PYQ — JEE Main 2022 (June 28 Shift 2)
Question: If the number of elements in the power set of a set $A$ is 64, then the number of elements in set $A$ is:
(a) 5
(b) 6
(c) 4
(d) 3
Solution:
Given $n(P(A)) = 64$.
Since $n(P(A)) = 2^{n(A)}$, we have:
Answer: (b) 6
Universal Set & Venn Diagrams
A universal set ($U$) is the base set containing all elements and of which all other sets under consideration are subsets. Venn diagrams are pictorial representations of sets using closed curves (usually circles) inside a rectangle representing the universal set.
5.1 Venn Diagram Representation
Venn Diagrams for Operations
- $A \cup B$: Total shaded area of both circles.
- $A \cap B$: The overlapping (intersection) region of both circles.
- $A - B$: The part of circle $A$ that does not overlap with $B$.
- $A'$ (Complement): The area inside the universal rectangle but outside circle $A$.
PYQ — JEE Main 2023 (Jan 30 Shift 1)
Question: In a Venn diagram, the overlapping region of two circles $A$ and $B$ represents which set?
(a) $A \cup B$
(b) $A \cap B$
(c) $A - B$
(d) $A \Delta B$
Solution:
The overlapping region contains elements that belong to both set $A$ and set $B$.
Answer: (b) $A \cap B$
Union of Sets
The union of two sets $A$ and $B$, denoted by $A \cup B$, is the set of elements that are in $A$, in $B$, or in both.
6.1 Definition and Properties
Most Repeated PYQ — Union Properties
- $A \cup A = A$ (Idempotent law)
- $A \cup \emptyset = A$ (Identity law)
- $A \cup U = U$ (Domination law)
- $A \cup B = B \cup A$ (Commutative law)
PYQ — JEE Main 2018 (Apr 9 Shift 1)
Question: If $A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$, find $A \cup B$.
Solution:
The union contains all distinct elements from both sets.
Answer: $\{1, 2, 3, 4, 5\}$
Intersection of Sets
The intersection of two sets $A$ and $B$, denoted by $A \cap B$, is the set of elements that are common to both $A$ and $B$.
7.1 Definition and Properties
Most Repeated PYQ — Intersection Properties
- $A \cap A = A$ (Idempotent law)
- $A \cap \emptyset = \emptyset$ (Identity law)
- $A \cap U = A$
- $A \cap B = B \cap A$ (Commutative law)
- If $A \subseteq B$, then $A \cap B = A$.
PYQ — JEE Main 2017 (May 3 Shift 2)
Question: If $A = \{x : x \text{ is a multiple of } 3\}$ and $B = \{x : x \text{ is a multiple of } 5\}$, then $A \cap B$ is:
(a) Multiples of 15
(b) Multiples of 8
(c) Multiples of 3
(d) Multiples of 5
Solution:
The intersection requires elements to be multiples of both 3 and 5.
LCM of 3 and 5 is 15.
Answer: (a)
Difference & Symmetric Difference
8.1 Difference of Sets
The difference $A - B$ is the set of elements in $A$ but not in $B$.
Similarly, $B - A = \{x : x \in B \text{ and } x \notin A\}$. Note that $A - B \neq B - A$.
8.2 Symmetric Difference
Most Repeated PYQ — Symmetric Difference
The symmetric difference of $A$ and $B$, denoted by $A \Delta B$, is the set of elements that belong to exactly one of $A$ or $B$.
PYQ — JEE Main 2020 (Oct 5 Shift 2)
Question: If $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$, find $A \Delta B$.
Solution:
$A \cup B = \{1, 2, 3, 4, 5, 6\}$
$A \cap B = \{3, 4\}$
$A \Delta B = (A \cup B) - (A \cap B)$
Answer: $\{1, 2, 5, 6\}$
Complement of a Set
The complement of a set $A$, denoted by $A'$ or $A^c$, is the set of all elements in the universal set $U$ that are not in $A$.
9.1 Properties of Complement
- $A \cup A' = U$
- $A \cap A' = \emptyset$
- $(A')' = A$ (Involution law)
- $\emptyset' = U$ and $U' = \emptyset$
PYQ — JEE Main 2016 (Apr 3 Shift 1)
Question: If $U = \{1, 2, ..., 10\}$ and $A = \{2, 4, 6, 8\}$, find $A'$.
Solution:
$A'$ contains elements in $U$ not in $A$.
Answer: $\{1, 3, 5, 7, 9, 10\}$
Algebra of Sets
Set operations follow several standard algebraic laws.
10.1 Commutative, Associative, and Distributive Laws
- Commutative: $A \cup B = B \cup A$ and $A \cap B = B \cap A$
- Associative: $(A \cup B) \cup C = A \cup (B \cup C)$ and $(A \cap B) \cap C = A \cap (B \cap C)$
- Distributive: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ and $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$
PYQ — JEE Main 2015 (Apr 2 Shift 1)
Question: For sets $A, B, C$, the expression $A \cup (B \cap C)$ is equal to:
(a) $(A \cup B) \cap (A \cup C)$
(b) $(A \cap B) \cup (A \cap C)$
(c) $(A \cup B) \cap C$
(d) $A \cap (B \cup C)$
Solution:
By the Distributive Law of union over intersection:
Answer: (a)
De Morgan's Laws
De Morgan's Laws are arguably the most frequently tested set theory property in JEE Mains. They relate union and intersection through complements.
11.1 The Laws
Most Repeated PYQ — De Morgan's Laws
1. Complement of a union is the intersection of complements:
2. Complement of an intersection is the union of complements:
These extend to any number of sets: $(A_1 \cup A_2 \cup ... \cup A_n)' = A_1' \cap A_2' \cap ... \cap A_n'$.
PYQ — JEE Main 2024 (Jan 30 Shift 2)
Question: Simplify $(A - B) \cup (A - C)$ in its simplest form, assuming $B \subseteq C$.
Solution:
$A - B = A \cap B'$
$A - C = A \cap C'$
Since $B \subseteq C$, we have $C' \subseteq B'$, so $B' \cup C' = B'$.
Answer: $A - B$
Cardinality of Two Sets
The cardinality of a set $A$, denoted by $n(A)$, is the number of elements in it.
12.1 Formula for Two Sets
Most Repeated PYQ — 2-Set Cardinality
- $n(A - B) = n(A \cup B) - n(B) = n(A) - n(A \cap B)$
- $n(A \Delta B) = n(A \cup B) - n(A \cap B) = n(A) + n(B) - 2n(A \cap B)$
- If $A$ and $B$ are disjoint, $n(A \cap B) = 0$, so $n(A \cup B) = n(A) + n(B)$.
PYQ — JEE Main 2023 (Apr 11 Shift 1)
Question: In a group of 50 students, 30 like cricket and 25 like football. If 10 like neither, how many like both?
Solution:
Total $n(U) = 50$. Let $C$ = cricket, $F$ = football.
$n(C \cup F) = 50 - 10 = 40$.
Using $n(C \cup F) = n(C) + n(F) - n(C \cap F)$:
Answer: 15
Cardinality of Three Sets
13.1 Formula for Three Sets
Most Repeated PYQ — 3-Set Cardinality
Exactly two sets:
Exactly one set:
PYQ — JEE Main 2024 (Apr 5 Shift 1)
Question: In a survey of 100 people, 60 read Hindi, 40 read English, and 30 read Marathi newspapers. 10 read both Hindi and English, 8 read both English and Marathi, and 5 read all three. If 12 read none, how many read both Hindi and Marathi?
Solution:
$n(H \cup E \cup M) = 100 - 12 = 88$.
Answer: 24
Important Results & Properties
A consolidated list of important results and properties.
14.1 Key Formulae Summary
Most Repeated PYQ — Key Formulae
1. Subset Count: $2^n$
2. Power Set: $n(P(A)) = 2^n$
3. Difference: $A - B = A \cap B'$
4. Symmetric Difference: $A \Delta B = (A \cup B) - (A \cap B)$
5. De Morgan 1: $(A \cup B)' = A' \cap B'$
6. De Morgan 2: $(A \cap B)' = A' \cup B'$
7. 2-Set Cardinality: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$
8. 3-Set Cardinality: $n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$
PYQ — JEE Main 2022 (July 27 Shift 1)
Question: If $A$ and $B$ are two sets such that $A \cup B = A \cap B$, then:
(a) $A \subset B$
(b) $B \subset A$
(c) $A = B$
(d) $A$ and $B$ are disjoint
Solution:
$A \cup B$ always contains $A \cap B$.
For them to be equal, no unique elements can exist in $A$ or $B$. Thus $A = B$.
Answer: (c) $A = B$
Previous Year Questions Analysis
JEE Main (Last 10 Years)
- ✓ Cardinality (3 sets): 30%
- ✓ Cardinality (2 sets): 20%
- ✓ De Morgan's Laws/Simplification: 20%
- ✓ Power Set/Subsets: 15%
- ✓ Operations on Sets: 15%
JEE Advanced (Last 10 Years)
- ✓ Multi-concept Set Problems: 40%
- ✓ Symmetric Difference: 25%
- ✓ De Morgan's Proofs: 20%
- ✓ Cardinality / Venn Logic: 15%
Top 10 Most Repeated Question Types
- 3-set cardinality word problems (Tea/Coffee/Milk, Newspapers)
- 2-set cardinality word problems
- Simplification using De Morgan's laws
- Finding Power Set cardinality
- Number of subsets/proper subsets
- Symmetric difference calculation
- Identifying equal, equivalent, or empty sets
- Applying Distributive Law
- Converting Roster to Set-Builder form
- Finding elements based on conditions (e.g., $x^2 = k$)
Weightage Analysis
Practice Problem Set
Level 1: Basic (JEE Main Standard)
- Write the set $\{x : x \in \mathbb{N} \text{ and } x^2 \le 10\}$ in roster form.
- Find the number of subsets of the set of vowels in the English alphabet.
- If $A = \{1, 3, 5\}$ and $B = \{2, 3\}$, find $A \Delta B$.
- Simplify $(A \cap B) \cup (A \cap B')$.
- In a group of 40 people, 25 like tea and 20 like coffee. If 5 like neither, how many like both?
Level 2: Intermediate (JEE Main/Advanced)
- Prove that $A - (B \cap C) = (A - B) \cup (A - C)$.
- In a survey of 60 people, 25 read H1, 26 read H2, and 26 read H3. 9 read both H1 and H3; 11 read both H2 and H3; 8 read all three. How many read both H1 and H2?
- Find $n(P(P(\emptyset)))$.
- Let $U = \{1, 2, ..., 20\}$. $A$ = even numbers, $B$ = multiples of 3. Find $n(A' \cap B)$.
> FAQ
Frequently Asked Questions
What is a Power Set?
The power set of a set A is the set of all subsets of A, including the empty set and A itself. If a set has n elements, its power set contains 2^n elements.
What are De Morgan's Laws in set theory?
De Morgan's Laws state that: 1) (A ∪ B)' = A' ∩ B' and 2) (A ∩ B)' = A' ∪ B'. These laws relate the union and intersection of sets to their complements.
What is the formula for n(A ∪ B ∪ C)?
The formula for the cardinality of the union of three sets is: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(A ∩ C) + n(A ∩ B ∩ C).
How many questions come from Sets in JEE Mains?
Typically, 1 question (4 marks) appears from Sets and Relations in every JEE Main session. It is often a quick, formula-based question testing cardinality rules or De Morgan's laws.