🔹 1. Introduction

A set is a well-defined collection of distinct objects.

👉 These objects are called elements or members of the set.

Example:

A = {1, 2, 3}
B = {a, b, c}

🔹 2. Representation of Sets

✔️ 1. Roster (Tabular) Form

List all elements inside braces.

Example:
A = {1, 2, 3, 4}

✔️ 2. Set-Builder Form

Describe elements using a property.

Example:
A = {x | x is a natural number less than 5}

🔹 3. Types of Sets

✔️ 1. Empty Set (Null Set)

Contains no elements.

Notation: ∅

✔️ 2. Finite Set

Contains limited elements.

✔️ 3. Infinite Set

Contains unlimited elements.

✔️ 4. Singleton Set

Contains exactly one element.

✔️ 5. Equal Sets

Two sets are equal if they have same elements.

✔️ 6. Equivalent Sets

Have same number of elements.

🔹 4. Subsets

✔️ Definition

A ⊆ B if every element of A is in B.

✔️ Important Results

Every set is subset of itself
∅ is subset of every set

✔️ Number of Subsets

$2^n$ 2n

(where n = number of elements)

🔹 5. Power Set

Set of all subsets of a set.

Example:
If A = {1,2}

Power set = {∅, {1}, {2}, {1,2}}

🔹 6. Universal Set

Set containing all elements under consideration.

Notation: U

🔹 7. Venn Diagrams (Concept)

Used to represent sets visually (not included here due to text-only format).

🔹 8. Operations on Sets

✔️ 1. Union

All elements in A or B:

$A \cup B$ A∪B

✔️ 2. Intersection

Common elements:

$A \cap B$ A∩B

✔️ 3. Difference

Elements in A but not in B:

$A – B$ A−B

✔️ 4. Complement

Elements not in A:

$A’ = U – A$ A′=U−A

🔹 9. Algebra of Sets (Important Laws)

✔️ Commutative Laws

$A \cup B = B \cup A$ A∪B=B∪A

$A \cap B = B \cap A$ A∩B=B∩A

✔️ Associative Laws

$(A \cup B) \cup C = A \cup (B \cup C)$ (A∪B)∪C=A∪(B∪C)

✔️ Distributive Laws

$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ A∩(B∪C)=(A∩B)∪(A∩C)

✔️ De Morgan’s Laws

$(A \cup B)’ = A’ \cap B’$ (A∪B)′=A′∩B′

$(A \cap B)’ = A’ \cup B’$ (A∩B)′=A′∪B′

🔹 10. Important Formulas

✔️ Number of Elements in Union

$n(A \cup B) = n(A) + n(B) – n(A \cap B)$ n(A∪B)=n(A)+n(B)−n(A∩B)

$P(A)$ P(A)

$P(B)$ P(B)

$P(A\cap B)$ P(A∩B)

$P(A\cup B)=P(A)+P(B)-P(A\cap B)\approx 0.80$ P(A∪B)=P(A)+P(B)−P(A∩B)≈0.80AB0.20

✔️ For Three Sets

$n(A \cup B \cup C) = n(A)+n(B)+n(C) – n(A\cap B) – n(B\cap C) – n(A\cap C) + n(A\cap B\cap C)$ n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(A∩C)+n(A∩B∩C)

🔹 11. Important Results

✔️ A ∪ A = A
✔️ A ∩ A = A
✔️ A ∪ ∅ = A
✔️ A ∩ ∅ = ∅

🔹 12. Applications

✔️ Probability
✔️ Logic
✔️ Computer science
✔️ Data analysis

🔹 13. JEE & CBSE Important Points

✔️ Practice Venn-based questions
✔️ Inclusion-exclusion principle important
✔️ Set identities frequently asked
✔️ Cardinality problems common

🔹 14. Common Mistakes

❌ Confusing union and intersection
❌ Wrong use of complement
❌ Calculation errors in formulas
❌ Ignoring universal set

🔹 15. Practice Questions

Find subsets of a set
Solve union/intersection problems
Apply De Morgan’s laws
Solve cardinality problems
Use inclusion-exclusion principle

🔹 16. Quick Revision Sheet

n subsets = 2ⁿ
A ∪ B, A ∩ B
A’ = U − A
De Morgan’s laws
n(A ∪ B) formula

🔹 17. Conclusion

Sets is a basic and very important chapter that forms the foundation for:

Relations & Functions
Probability
Logic

👉 Mastering this chapter ensures strong mathematical understanding and better performance in CBSE & JEE exams.

Sets | Class 11 Maths Notes & Venn Diagrams