What is a set in mathematics?

A set is a well-defined collection of distinct objects, called elements or members. Sets are typically denoted by capital letters and elements are enclosed in curly braces. For example, A = {1, 2, 3, 4, 5}.

What is De Morgan's Law in set theory?

De Morgan's Laws state that: (1) (A ∪ B)' = A' ∩ B' - The complement of union is the intersection of complements. (2) (A ∩ B)' = A' ∪ B' - The complement of intersection is the union of complements.

What is the difference between subset and proper subset?

A is a subset of B (A ⊆ B) if all elements of A are in B. A is a proper subset of B (A ⊂ B) if all elements of A are in B but A ≠ B. Every set is a subset of itself but not a proper subset of itself.

How to find the number of subsets of a set?

If a set has n elements, then the total number of subsets is 2^n and the number of proper subsets is 2^n - 1. For example, if set A has 3 elements, it has 2³ = 8 subsets and 7 proper subsets.

The power set of a set A, denoted by P(A), is the set of all subsets of A including the empty set and A itself. If |A| = n, then |P(A)| = 2^n.

What is the formula for n(A ∪ B ∪ C)?

For three sets A, B, and C: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(C ∩ A) + n(A ∩ B ∩ C). This is the inclusion-exclusion principle for three sets.

Sets for JEE 2025-26 | Complete Notes with Formulas, Venn Diagrams & Solved Examples

Set Fundamentals

A set is a well-defined collection of distinct objects. The objects in a set are called elements or members of the set. Sets form the foundation of modern mathematics and are essential for JEE preparation.

1.1 What is a Set?

Definition

A set is a well-defined collection of distinct objects. "Well-defined" means that for any given object, we can definitely say whether it belongs to the set or not.

Examples of Well-Defined Sets:

The set of natural numbers less than 10
The set of vowels in English alphabet
The set of prime numbers less than 20

NOT Well-Defined (Not Sets):

The collection of beautiful flowers (subjective)
The collection of intelligent students (ambiguous)
The collection of good teachers (not well-defined)

1.2 Representation of Sets

1. Roster Form (Tabular Form)

Elements are listed within curly braces { }, separated by commas.

A = {1, 2, 3, 4, 5}

B = {a, e, i, o, u}

C = {2, 4, 6, 8, ...}

Note: Order doesn't matter. {1, 2, 3} = {3, 2, 1}

2. Set-Builder Form

Elements are described by a property or rule.

A = {x : x ∈ ℕ, x ≤ 5}

B = {x : x is a vowel}

C = {x : x = 2n, n ∈ ℕ}

Read as: "The set of all x such that..."

📝 Solved Example 1

Question: Express the following sets in both roster form and set-builder form:
(a) Set of all odd natural numbers less than 10
(b) Set of all prime numbers less than 15

Solution:

(a) Odd natural numbers less than 10:

Roster Form: A = {1, 3, 5, 7, 9}

Set-Builder Form: A = {x : x ∈ ℕ, x < 10, x is odd}

or A = {x : x = 2n - 1, n ∈ ℕ, n ≤ 5}

(b) Prime numbers less than 15:

Roster Form: B = {2, 3, 5, 7, 11, 13}

Set-Builder Form: B = {x : x is prime, x < 15}

1.3 Set Notation and Symbols

Symbol	Meaning	Example
∈	belongs to, is an element of	3 ∈ {1, 2, 3, 4}
∉	does not belong to	5 ∉ {1, 2, 3, 4}
⊆	is a subset of	{1, 2} ⊆ {1, 2, 3}
⊂	is a proper subset of	{1, 2} ⊂ {1, 2, 3}
⊇	is a superset of	{1, 2, 3} ⊇ {1, 2}
∅ or { }	empty set, null set	{x : x² = -1, x ∈ ℝ} = ∅
ℕ	set of natural numbers	{1, 2, 3, 4, ...}
ℤ	set of integers	{..., -2, -1, 0, 1, 2, ...}
ℚ	set of rational numbers	{p/q : p, q ∈ ℤ, q ≠ 0}
ℝ	set of real numbers	All rational and irrational numbers

💡 Important Points to Remember

Elements are distinct: {1, 2, 2, 3} = {1, 2, 3} (repetition not allowed)
Order doesn't matter: {1, 2, 3} = {3, 2, 1} = {2, 1, 3}
Notation: Use 'a ∈ A' not 'a ⊆ A' (element vs subset)
Empty set: ∅ is different from {∅} (first has 0 elements, second has 1)
Natural numbers: Some books include 0, some don't. Check exam pattern!

Types of Sets

Sets are classified based on the number of elements they contain, their relationship with other sets, or special properties. Understanding these types is crucial for solving JEE problems efficiently.

2.1 Classification Based on Number of Elements

1. Finite Set

A set with countable number of elements.

Examples:

A = {1, 2, 3, 4, 5}

B = {x : x ∈ ℕ, x ≤ 100}

C = {vowels in English}

n(A) = 5, n(B) = 100, n(C) = 5

2. Infinite Set

A set with uncountable number of elements.

Examples:

ℕ = {1, 2, 3, 4, ...}

ℤ = {..., -2, -1, 0, 1, 2, ...}

A = {x : x ∈ ℝ, 0 < x < 1}

n(ℕ) = ∞, n(ℤ) = ∞, n(A) = ∞

3. Empty Set (Null Set)

A set with no elements. Denoted by ∅ or { }.

Examples:

A = {x : x² = -1, x ∈ ℝ}

B = {x : x ∈ ℕ, 3 < x < 4}

C = {x : x ≠ x}

Note: {0} ≠ ∅ (has one element)

4. Singleton Set

A set with exactly one element.

Examples:

A = {0}

B = {x : x + 5 = 8}

C = {x : x² = 16, x > 0}

n(A) = 1, n(B) = 1, n(C) = 1

2.2 Special Types of Sets

Universal Set (U or ξ)

The set containing all objects under consideration for a particular discussion. All other sets are subsets of the universal set.

Example:

If we're studying natural numbers less than 20:

U = {1, 2, 3, 4, ..., 19}

Then we can define:

A = {even numbers} = {2, 4, 6, ..., 18}

B = {prime numbers} = {2, 3, 5, 7, 11, 13, 17, 19}

Equal Sets

Two sets A and B are equal if they have exactly the same elements.

A = B ⟺ (A ⊆ B and B ⊆ A)

Example:

{1, 2, 3} = {3, 2, 1}

{a, b} = {b, a}

Equivalent Sets

Two sets having same number of elements (same cardinality).

A ~ B ⟺ n(A) = n(B)

Example:

{1, 2, 3} ~ {a, b, c}

{x, y} ~ {5, 10}

Disjoint Sets

Two sets with no common elements.

A ∩ B = ∅

Example:

A = {1, 2, 3}

B = {4, 5, 6}

A ∩ B = ∅

Overlapping Sets

Two sets with at least one common element.

A ∩ B ≠ ∅

Example:

A = {1, 2, 3, 4}

B = {3, 4, 5, 6}

A ∩ B = {3, 4}

📝 Solved Example 2

Question: Classify the following sets as finite, infinite, empty, or singleton:
(a) A = {x : x ∈ ℕ, x² = 25}
(b) B = {x : x ∈ ℤ, -5 < x < 5}
(c) C = {x : x ∈ ℝ, x² + 1 = 0}
(d) D = {x : x is a multiple of 5}

Solution:

(a) A = {x : x ∈ ℕ, x² = 25}

x² = 25 ⟹ x = ±5

Since x ∈ ℕ (natural numbers), only x = 5

A = {5}

Answer: Singleton Set

(b) B = {x : x ∈ ℤ, -5 < x < 5}

Integers between -5 and 5 (not including -5 and 5)

B = {-4, -3, -2, -1, 0, 1, 2, 3, 4}

n(B) = 9

Answer: Finite Set

(c) C = {x : x ∈ ℝ, x² + 1 = 0}

x² + 1 = 0 ⟹ x² = -1

No real number satisfies this equation

C = ∅

Answer: Empty Set

(d) D = {x : x is a multiple of 5}

Multiples of 5 are: ..., -10, -5, 0, 5, 10, 15, ...

D = {..., -10, -5, 0, 5, 10, 15, ...}

Answer: Infinite Set

⚠️ Common Mistakes to Avoid

{0} is NOT an empty set - it's a singleton set containing 0
∅ ≠ {∅} - first is empty, second contains empty set as element
Equal sets vs Equivalent sets - Equal means same elements, Equivalent means same count
Proper subset: A ⊂ B means A is subset but A ≠ B
Subset: Every set is a subset of itself, but NOT a proper subset of itself

Set Operations

Set operations allow us to combine, compare, and manipulate sets. The four fundamental operations are Union, Intersection, Difference, and Complement. These form the basis of solving complex Venn diagram problems in JEE.

3.1 Union of Sets (∪)

Definition

The union of sets A and B is the set of all elements that are in A or in B or in both.

A \cup B = \{x : x \in A \text{ or } x \in B\}

Example:

A = {1, 2, 3, 4}

B = {3, 4, 5, 6}

A ∪ B = {1, 2, 3, 4, 5, 6}

Note: Common elements (3, 4) appear only once

3.2 Intersection of Sets (∩)

Definition

The intersection of sets A and B is the set of all elements that are in both A and B.

A \cap B = \{x : x \in A \text{ and } x \in B\}

Example:

A = {1, 2, 3, 4}

B = {3, 4, 5, 6}

A ∩ B = {3, 4}

Only common elements

3.3 Difference of Sets (A - B or A \ B)

Definition

The difference A - B is the set of all elements that are in A but not in B.

A - B = \{x : x \in A \text{ and } x \notin B\}

Example:

A = {1, 2, 3, 4}

B = {3, 4, 5, 6}

A - B = {1, 2}

B - A = {5, 6}

Note: A - B ≠ B - A (not commutative)

3.4 Complement of a Set (A' or A^c)

Definition

The complement of set A (with respect to universal set U) is the set of all elements in U that are not in A.

A' = U - A = \{x : x \in U \text{ and } x \notin A\}

Example:

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {2, 4, 6, 8, 10} (even numbers)

A' = {1, 3, 5, 7, 9} (odd numbers)

3.5 Properties of Set Operations

Property	Union (∪)	Intersection (∩)
Commutative	A ∪ B = B ∪ A	A ∩ B = B ∩ A
Associative	(A ∪ B) ∪ C = A ∪ (B ∪ C)	(A ∩ B) ∩ C = A ∩ (B ∩ C)
Identity	A ∪ ∅ = A	A ∩ U = A
Idempotent	A ∪ A = A	A ∩ A = A
Distributive	A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)	A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Domination	A ∪ U = U	A ∩ ∅ = ∅
Complement	A ∪ A' = U	A ∩ A' = ∅

📝 Solved Example 3

Question: If A = {1, 2, 3, 4, 5}, B = {2, 4, 6, 8}, and C = {3, 4, 5, 6}, find:
(a) A ∪ B ∪ C
(b) A ∩ B ∩ C
(c) (A ∪ B) ∩ C
(d) A - (B ∩ C)

Solution:

(a) A ∪ B ∪ C

Combine all elements from A, B, and C (no repetition)

A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 8}

(b) A ∩ B ∩ C

A ∩ B = {2, 4}

(A ∩ B) ∩ C = {2, 4} ∩ {3, 4, 5, 6}

A ∩ B ∩ C = {4}

(c) (A ∪ B) ∩ C

A ∪ B = {1, 2, 3, 4, 5, 6, 8}

(A ∪ B) ∩ C = {1, 2, 3, 4, 5, 6, 8} ∩ {3, 4, 5, 6}

(A ∪ B) ∩ C = {3, 4, 5, 6}

(d) A - (B ∩ C)

B ∩ C = {2, 4, 6, 8} ∩ {3, 4, 5, 6} = {4, 6}

A - (B ∩ C) = {1, 2, 3, 4, 5} - {4, 6}

A - (B ∩ C) = {1, 2, 3, 5}

💡 JEE Quick Tips

Union (∪): Think "OR" - takes all elements from both sets
Intersection (∩): Think "AND" - takes only common elements
Difference (A - B): Elements in A but not in B
Symmetric Difference: (A - B) ∪ (B - A) = (A ∪ B) - (A ∩ B)
De Morgan's Laws: (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'

Venn Diagrams

Venn diagrams are visual representations of sets using circles or closed curves. They are extremely useful for solving complex set problems in JEE and make abstract concepts concrete. Mastering Venn diagrams can save 2-3 minutes per question!

4.1 Basic Venn Diagrams for Two Sets

A ∪ B (Union)

Shaded region represents A ∪ B

All of A + All of B

A ∩ B (Intersection)

Shaded region represents A ∩ B

Common elements only

A - B (Difference)

Shaded region represents A - B

In A but not in B

A' (Complement)

Shaded region represents A'

In U but not in A

4.2 Venn Diagrams for Three Sets

Three Sets A, B, C with All Regions

Region Descriptions:

①: Only A (A - B - C)

②: Only B (B - A - C)

③: Only C (C - A - B)

④: A ∩ B only (not C)

⑤: B ∩ C only (not A)

⑥: A ∩ C only (not B)

⑦: (Region outside all 3)

⑧: A ∩ B ∩ C (all three)

4.3 Cardinality Formulas Using Venn Diagrams

For Two Sets:

Union Formula (Most Important for JEE):

n(A \cup B) = n(A) + n(B) - n(A \cap B)

We subtract n(A ∩ B) because it's counted twice in n(A) + n(B)

Disjoint Sets (when A ∩ B = ∅):

n(A \cup B) = n(A) + n(B)

Difference Formula:

n(A - B) = n(A) - n(A \cap B)

Symmetric Difference:

n(A \triangle B) = n(A) + n(B) - 2n(A \cap B)

For Three Sets (JEE Advanced Level):

Inclusion-Exclusion Principle:

n(A \cup B \cup C) = n(A) + n(B) + n(C)\] \[- n(A \cap B) - n(B \cap C) - n(C \cap A)\] \[+ n(A \cap B \cap C)

Step-by-Step Logic:

Add all individual sets: n(A) + n(B) + n(C)
Subtract pairwise intersections (counted twice): - n(A∩B) - n(B∩C) - n(A∩C)
Add back triple intersection (subtracted thrice, should be subtracted twice): + n(A∩B∩C)

📝 Solved Example 4 (JEE Main 2023 Pattern)

Question: In a class of 100 students, 60 play cricket, 50 play football, and 40 play both. Find:
(a) Number of students who play at least one game
(b) Number of students who play exactly one game
(c) Number of students who play neither game

Solution:

Given:

Total students = 100
n(Cricket) = C = 60
n(Football) = F = 50
n(C ∩ F) = 40

(a) Students playing at least one game = n(C ∪ F)

n(C \cup F) = n(C) + n(F) - n(C \cap F)

\[= 60 + 50 - 40\]

= 70 \text{ students}

(b) Students playing exactly one game:

Only Cricket = n(C) - n(C ∩ F) = 60 - 40 = 20

Only Football = n(F) - n(C ∩ F) = 50 - 40 = 10

\text{Total} = 20 + 10 = 30 \text{ students}

(c) Students playing neither game:

= \text{Total} - n(C \cup F)

\[= 100 - 70\]

= 30 \text{ students}

📝 Solved Example 5 (JEE Advanced Pattern - 3 Sets)

Question: In a survey of 200 students:
• 120 study Mathematics
• 90 study Physics
• 70 study Chemistry
• 40 study Math and Physics
• 30 study Physics and Chemistry
• 50 study Math and Chemistry
• 20 study all three subjects
Find: (a) Students studying at least one subject (b) Students studying exactly two subjects

Solution:

Given:

n(M) = 120, n(P) = 90, n(C) = 70

n(M ∩ P) = 40, n(P ∩ C) = 30, n(M ∩ C) = 50

n(M ∩ P ∩ C) = 20

(a) Using Inclusion-Exclusion Principle:

n(M \cup P \cup C) = n(M) + n(P) + n(C)

- n(M \cap P) - n(P \cap C) - n(M \cap C)

+ n(M \cap P \cap C)

\[= 120 + 90 + 70 - 40 - 30 - 50 + 20\]

= 180 \text{ students}

(b) Students studying exactly two subjects:

Only M and P (not C) = n(M ∩ P) - n(M ∩ P ∩ C) = 40 - 20 = 20

Only P and C (not M) = n(P ∩ C) - n(M ∩ P ∩ C) = 30 - 20 = 10

Only M and C (not P) = n(M ∩ C) - n(M ∩ P ∩ C) = 50 - 20 = 30

\text{Total} = 20 + 10 + 30 = 60 \text{ students}

🎯 Venn Diagram Shortcuts for JEE

Always start from the innermost region: For 3 sets, fill A ∩ B ∩ C first
Work outwards: Then fill pairwise intersections, then individual sets
Maximum regions: 2 sets → 4 regions, 3 sets → 8 regions, n sets → 2ⁿ regions
Quick check: Sum of all regions = Total number of elements
For "only A": Subtract all intersections from n(A)
Use variables: If numbers aren't given, use x, y, z for unknown regions

⚠️ Common Venn Diagram Mistakes

Double counting: Forgetting to subtract intersections in union formula
"At least one" ≠ "Exactly one": At least one = Union, Exactly one = Symmetric difference
Negative values: If you get negative in a region, check your calculations
Total doesn't match: Sum of all regions must equal total given
Three set formula: Don't forget the +n(A∩B∩C) term at the end

De Morgan's Laws

De Morgan's Laws are fundamental theorems in set theory that relate the complement of unions and intersections. These laws are extensively used in JEE to simplify complex set expressions and prove set identities.

5.1 The Two Laws

First Law

(A \cup B)' = A' \cap B'

The complement of a UNION is the INTERSECTION of complements

In words:

"NOT (A OR B)" is the same as "(NOT A) AND (NOT B)"

Second Law

(A \cap B)' = A' \cup B'

The complement of an INTERSECTION is the UNION of complements

In words:

"NOT (A AND B)" is the same as "(NOT A) OR (NOT B)"

💡 Easy Way to Remember

When you take complement of a set expression:

Union (∪) becomes Intersection (∩)
Intersection (∩) becomes Union (∪)
Each set gets complemented

∪ ⟷ ∩ (Operations swap)

5.2 Extended De Morgan's Laws (For Multiple Sets)

For Three Sets:

(A \cup B \cup C)' = A' \cap B' \cap C'

(A \cap B \cap C)' = A' \cup B' \cup C'

General Form (n sets):

\left(\bigcup_{i=1}^{n} A_i\right)' = \bigcap_{i=1}^{n} A_i'

\left(\bigcap_{i=1}^{n} A_i\right)' = \bigcup_{i=1}^{n} A_i'

5.3 Proof of De Morgan's Laws

📐 Proof of First Law: (A ∪ B)' = A' ∩ B'

Method 1: Element-wise Proof

To Prove: (A ∪ B)' = A' ∩ B'

Part 1: Show (A ∪ B)' ⊆ A' ∩ B'

Let x ∈ (A ∪ B)'

⟹ x ∉ (A ∪ B)

⟹ x ∉ A and x ∉ B

⟹ x ∈ A' and x ∈ B'

⟹ x ∈ A' ∩ B'

∴ (A ∪ B)' ⊆ A' ∩ B'

Part 2: Show A' ∩ B' ⊆ (A ∪ B)'

Let x ∈ A' ∩ B'

⟹ x ∈ A' and x ∈ B'

⟹ x ∉ A and x ∉ B

⟹ x ∉ (A ∪ B)

⟹ x ∈ (A ∪ B)'

∴ A' ∩ B' ⊆ (A ∪ B)'

∴ (A ∪ B)' = A' ∩ B' (Proved)

5.4 Applications of De Morgan's Laws

📝 Solved Example 6

Question: Simplify: [(A ∪ B)' ∪ (A ∩ B)']'

Solution:

Step-by-step simplification:

Step 1: Apply De Morgan's Law to inner expressions

(A ∪ B)' = A' ∩ B'

(A ∩ B)' = A' ∪ B'

Step 2: Substitute

[(A' ∩ B') ∪ (A' ∪ B')]'

Step 3: Apply De Morgan's Law to outer expression

= (A' ∩ B')' ∩ (A' ∪ B')'

Step 4: Apply De Morgan's Law again

= (A ∪ B) ∩ (A ∩ B)

Step 5: Since (A ∩ B) ⊆ (A ∪ B)

= A ∩ B

\text{Final Answer: } A \cap B

📝 Solved Example 7 (JEE Advanced Pattern)

Question: If U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, verify De Morgan's Law: (A ∪ B)' = A' ∩ B'

Solution:

LHS: (A ∪ B)'

A ∪ B = {1, 2, 3, 4} ∪ {3, 4, 5, 6}

A ∪ B = {1, 2, 3, 4, 5, 6}

(A ∪ B)' = U - (A ∪ B)

(A ∪ B)' = {7, 8}

RHS: A' ∩ B'

A' = U - A = {5, 6, 7, 8}

B' = U - B = {1, 2, 7, 8}

A' ∩ B' = {5, 6, 7, 8} ∩ {1, 2, 7, 8}

A' ∩ B' = {7, 8}

LHS = RHS = {7, 8}

∴ De Morgan's Law Verified ✓

5.5 Important Set Identities Using De Morgan's Laws

Key Identities to Remember for JEE:

(A')' = A

Complement of complement

U' = ∅, ∅' = U

Universal and empty set

A - B = A ∩ B'

Set difference

(A - B)' = A' ∪ B

Complement of difference

A ∪ A' = U

Law of excluded middle

A ∩ A' = ∅

Law of contradiction

⚠️ Common Mistakes with De Morgan's Laws

Wrong: (A ∪ B)' = A ∪ B' (forgot to change ∪ to ∩)
Wrong: (A ∩ B)' = A ∩ B' (forgot to change ∩ to ∪)
Wrong: (A ∪ B)' = A' ∪ B' (complemented sets but didn't swap operation)
Remember: Both operations AND sets change when taking complement
Practice: Write out each step - don't do it all mentally in exam

🎯 JEE Strategy for De Morgan's Laws

Identify the outermost operation - Apply De Morgan's from outside to inside
Use Venn diagrams - For verification, draw quick diagrams
Memorize the pattern: ∪ ↔ ∩ (operations swap), A → A' (sets complement)
For complex expressions: Work step by step, don't skip steps in exam
Quick check: If you get (A ∪ B)' = (A ∩ B)' something is wrong!

Power Sets

The power set is one of the most important concepts in advanced set theory and appears frequently in JEE Advanced. Understanding power sets is crucial for counting problems and advanced set operations.

6.1 Definition of Power Set

Power Set P(A)

The power set of a set A, denoted by P(A) or 2^A, is the set of all possible subsets of A, including the empty set ∅ and A itself.

P(A) = \{B : B \subseteq A\}

Important Formula:

\text{If } |A| = n, \text{ then } |P(A)| = 2^n

Where |A| denotes the number of elements (cardinality) of set A

📝 Solved Example 8 - Finding Power Sets

Question: Find the power set of:
(a) A = {1, 2}
(b) B = {a, b, c}
(c) C = ∅

Solution:

(a) A = {1, 2}

n(A) = 2, so |P(A)| = 2² = 4 subsets

All subsets:

∅ (empty set - 0 elements)
{1} (1 element)
{2} (1 element)
{1, 2} (2 elements - set itself)

P(A) = {∅, {1}, {2}, {1, 2}}

(b) B = {a, b, c}

n(B) = 3, so |P(B)| = 2³ = 8 subsets

All subsets:

0 elements:

∅

1 element:

{a}, {b}, {c}

2 elements:

{a,b}, {b,c}, {a,c}

3 elements:

{a, b, c}

P(B) = {∅, {a}, {b}, {c}, {a,b}, {b,c}, {a,c}, {a,b,c}}

(c) C = ∅ (Empty Set)

n(C) = 0, so |P(C)| = 2⁰ = 1 subset

The only subset of empty set is empty set itself

P(∅) = {∅}

Note: P(∅) ≠ ∅ because P(∅) has one element!

6.2 Properties of Power Sets

Important Properties:

1. Cardinality of Power Set:

\text{If } |A| = n, \text{ then } |P(A)| = 2^n

2. Empty Set is Always a Member:

∅ ∈ P(A) for any set A

3. Set is Member of its Power Set:

A ∈ P(A) for any set A

4. Subset Relationship:

A \subseteq B \implies P(A) \subseteq P(B)

5. Power Set of Union:

P(A) \cup P(B) \subseteq P(A \cup B)

Note: Equality doesn't always hold

6. Power Set of Intersection:

P(A \cap B) = P(A) \cap P(B)

6.3 Number of Subsets with Specific Sizes

Combinatorial Formulas (Very Important for JEE):

Number of subsets with exactly r elements:

\binom{n}{r} = \frac{n!}{r!(n-r)!}

This is the binomial coefficient "n choose r"

Total number of subsets (all sizes):

\sum_{r=0}^{n} \binom{n}{r} = 2^n

Number of proper subsets:

\[2^n - 1\]

(Exclude the set itself)

Number of non-empty subsets:

\[2^n - 1\]

(Exclude the empty set)

📝 Solved Example 9 (JEE Main Pattern)

Question: A set A has 5 elements.
(a) Find the total number of subsets
(b) How many subsets have exactly 3 elements?
(c) How many proper subsets does A have?
(d) How many non-empty subsets does A have?

Solution:

Given: n(A) = 5

(a) Total number of subsets = |P(A)|

\[|P(A)| = 2^n = 2^5\]

= 32 subsets

(b) Subsets with exactly 3 elements:

Use combination formula:

\binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5!}{3! \cdot 2!}

= \frac{5 \times 4 \times 3!}{3! \times 2 \times 1} = \frac{20}{2}

= 10 subsets

(c) Number of proper subsets:

Proper subsets = Total subsets - 1 (exclude A itself)

\[= 2^5 - 1 = 32 - 1\]

= 31 proper subsets

(d) Number of non-empty subsets:

Non-empty subsets = Total subsets - 1 (exclude ∅)

\[= 2^5 - 1 = 32 - 1\]

= 31 non-empty subsets

💡 Quick Counting Tricks for Power Sets

What to Count	Formula	For n=5
Total subsets	2ⁿ	32
Proper subsets	2ⁿ - 1	31
Non-empty subsets	2ⁿ - 1	31
Non-empty proper subsets	2ⁿ - 2	30
Subsets with r elements	ⁿCᵣ	⁵C₃ = 10

⚠️ Common Power Set Mistakes

P(∅) ≠ ∅: P(∅) = {∅} which has one element!
∅ vs {∅}: ∅ is empty, {∅} contains empty set as element
Elements vs Subsets: 1 ∈ A but {1} ⊆ A (different relations)
Notation: Elements of power set are SETS, not individual elements
Counting: Don't forget to include ∅ and A itself in power set

📝 Previous Year Questions Analysis

JEE Main (Last 5 Years)

✓ Venn Diagrams & Cardinality: 45%
✓ Set Operations: 30%
✓ Power Sets & Subsets: 15%
✓ De Morgan's Laws: 10%

JEE Advanced (Last 5 Years)

✓ Complex Venn diagrams (3 sets): 40%
✓ Set identities & proofs: 30%
✓ Power sets & counting: 20%
✓ Mixed concepts: 10%

Weightage Analysis

JEE Main: 8-12 marks (2-3 questions)

JEE Advanced: 6-9 marks (1-2 questions)

Difficulty Level: Easy to Medium

Time Required for Prep: 3-4 hours

Download Complete PYQ PDF (2015-2024)

🎯 Practice Problem Set

Level 1: Basic (JEE Main Foundation)

If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find A ∪ B, A ∩ B, A - B, and B - A.
Write the power set of A = {a, b}.
If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {2, 4, 6, 8, 10}, find A'.
Verify De Morgan's law (A ∪ B)' = A' ∩ B' for A = {1, 2} and B = {2, 3}.
How many subsets does a set with 4 elements have?
Express {x : x² - 5x + 6 = 0} in roster form.
If n(A) = 40, n(B) = 30, n(A ∩ B) = 15, find n(A ∪ B).
List all subsets of {1, 2, 3} that contain exactly 2 elements.

Level 2: Intermediate (JEE Main Standard)

In a class of 100, 60 play hockey, 50 play cricket, 40 play both. How many play neither?
Simplify: (A ∪ B)' ∩ (A ∩ B')'
If |P(A)| = 128, find |A|.
Prove that (A - B) ∪ (B - A) = (A ∪ B) - (A ∩ B).
In a survey of 200 students: 120 like Math, 90 like Physics, 70 like Chemistry, 40 like Math & Physics, 30 like Physics & Chemistry, 50 like Math & Chemistry, 20 like all three. Find those who like at least one subject.
How many 3-element subsets can be formed from a 7-element set?
If A ⊆ B and |A| = 5, |B| = 8, how many elements are in B - A?
Verify: (A ∩ B ∩ C)' = A' ∪ B' ∪ C' for given sets.

Level 3: Advanced (JEE Advanced Pattern)

If n(A) = 3, n(B) = 6, and A ⊂ B, how many distinct ordered pairs (x, y) exist where x ∈ A and y ∈ B?
Prove using set algebra: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).
How many non-empty proper subsets does a set with n elements have?
If |A × B| = 12 and |A| = 3, find |P(B)|.
In a survey, ratio of people who like tea to coffee to both is 5:3:2. If 100 like at least one, how many like exactly one?
Simplify: [(A' ∪ B')' ∩ (A ∩ B)']'
If A has n elements, how many ordered pairs (X, Y) exist such that X ⊆ Y ⊆ A?
Prove: If A ∩ B = A ∩ C and A ∪ B = A ∪ C, then B = C.

Get Solutions PDF with Step-by-Step Working

Year	JEE Main	JEE Advanced	Topic Focus
2024	2 Questions (8 marks)	1 Question (4 marks)	Venn diagrams, De Morgan's laws
2023	3 Questions (12 marks)	1 Question (3 marks)	3-set problems, Power sets
2022	2 Questions (8 marks)	2 Questions (6 marks)	Set identities, Cardinality

📑 Quick Navigation - Sets

Sets JEE Main & Advanced 2025-26

Set Fundamentals

1.1 What is a Set?

Definition

1.2 Representation of Sets

1. Roster Form (Tabular Form)

2. Set-Builder Form

📝 Solved Example 1

1.3 Set Notation and Symbols

💡 Important Points to Remember

Types of Sets

2.1 Classification Based on Number of Elements

1. Finite Set

2. Infinite Set

3. Empty Set (Null Set)

4. Singleton Set

2.2 Special Types of Sets

Universal Set (U or ξ)

Equal Sets

Equivalent Sets

Disjoint Sets

Overlapping Sets

📝 Solved Example 2

⚠️ Common Mistakes to Avoid

Set Operations

3.1 Union of Sets (∪)

Definition

3.2 Intersection of Sets (∩)

Definition

3.3 Difference of Sets (A - B or A \ B)

Definition

3.4 Complement of a Set (A' or Ac)

Definition

3.5 Properties of Set Operations

📝 Solved Example 3

💡 JEE Quick Tips

Venn Diagrams

4.1 Basic Venn Diagrams for Two Sets

A ∪ B (Union)

A ∩ B (Intersection)

A - B (Difference)

A' (Complement)

4.2 Venn Diagrams for Three Sets

Three Sets A, B, C with All Regions

Region Descriptions:

4.3 Cardinality Formulas Using Venn Diagrams

For Two Sets:

Union Formula (Most Important for JEE):

Disjoint Sets (when A ∩ B = ∅):

Difference Formula:

Symmetric Difference:

For Three Sets (JEE Advanced Level):

Inclusion-Exclusion Principle:

Step-by-Step Logic:

📝 Solved Example 4 (JEE Main 2023 Pattern)

📝 Solved Example 5 (JEE Advanced Pattern - 3 Sets)

🎯 Venn Diagram Shortcuts for JEE

⚠️ Common Venn Diagram Mistakes

De Morgan's Laws

5.1 The Two Laws

First Law

Second Law

💡 Easy Way to Remember

5.2 Extended De Morgan's Laws (For Multiple Sets)

For Three Sets:

General Form (n sets):

5.3 Proof of De Morgan's Laws

📐 Proof of First Law: (A ∪ B)' = A' ∩ B'

5.4 Applications of De Morgan's Laws

📝 Solved Example 6

📝 Solved Example 7 (JEE Advanced Pattern)

5.5 Important Set Identities Using De Morgan's Laws

Key Identities to Remember for JEE:

⚠️ Common Mistakes with De Morgan's Laws

🎯 JEE Strategy for De Morgan's Laws

Power Sets

6.1 Definition of Power Set

Power Set P(A)

📝 Solved Example 8 - Finding Power Sets

3.4 Complement of a Set (A' or A^c)