Set Operations & Laws Made Easy: Union, Intersection, Difference & De Morgan’s Law

Set operations are the core of set theory. They allow us to combine, compare, and manipulate sets to solve mathematical problems. From board exams to JEE, mastering set operations ensures quick and accurate answers.

In this guide, we’ll cover:

• Union, intersection, difference, complement of sets
• Venn diagrams for visualization
• Important laws of sets
• De Morgan’s laws with examples
• FAQs to clear common doubts

Union of Sets (A ∪ B)

The union of two sets A and B is the set containing all elements from both A and B, without duplication.

Symbol: A ∪ B

Definition: A ∪ B = {x | x ∈ A or x ∈ B}

Example:

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

A ∪ B = {1, 2, 3, 4, 5}

Venn Diagram: Two overlapping circles shaded completely.

Intersection of Sets (A ∩ B)

The intersection of sets A and B includes only the elements common to both.

Symbol: A ∩ B

Definition: A ∩ B = {x | x ∈ A and x ∈ B}

Example:

A = {2, 4, 6}, B = {4, 6, 8}

A ∩ B = {4, 6}

Venn Diagram: Only the overlapping area is shaded.

Difference of Sets (A – B)

The difference of sets is the part of A that does not belong to B.

Symbol: A – B

Definition: A – B = {x | x ∈ A and x ∉ B}

Example:

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

A – B = {1, 2}

Venn Diagram: Circle A shaded except for the overlapping portion with B.

Complement of a Set (A′)

The complement of set A (within universal set U) is everything in U that is not in A.

Symbol: A′

Definition: A′ = {x | x ∈ U and x ∉ A}

Example:

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

A′ = {2, 4, 5}

Laws of Sets (Class 11)

Law	Formula	Meaning
Idempotent Law	A ∪ A = A, A ∩ A = A	Repetition doesn’t change sets
Commutative Law	A ∪ B = B ∪ A, A ∩ B = B ∩ A	Order doesn’t matter
Associative Law	(A ∪ B) ∪ C = A ∪ (B ∪ C)	Grouping doesn’t matter
Distributive Law	A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)	Like algebra distribution
Identity Law	A ∪ Ø = A, A ∩ U = A	Null/Universal rules
Complement Law	A ∪ A′ = U, A ∩ A′ = Ø	Set & complement cover all
Double Complement	(A′)′ = A	Two negatives = original set

De Morgan’s Laws of Sets

These laws describe the relationship between complements of unions and intersections:

1. (A ∪ B)′ = A′∩ B′
2. (A ∩ B)′ = A′∪ B′

Example:

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

A = {1, 2}, B = {2, 3}

• A ∪ B = {1, 2, 3} → (A ∪ B)′ = {4, 5}
• A′ = {3, 4, 5}, B′ = {1, 4, 5} → A′∩ B′ = {4, 5} 

Why Important:

• De Morgan’s laws appear in board exams, JEE, and computer science (logic circuits, Boolean algebra).

Exam Tip

• Draw a Venn diagram for every set operation — it reduces mistakes.
• Memorize De Morgan’s laws using the phrase: “Union turns into intersection, intersection turns into union when complemented.”