Intersection of Sets in Discrete Mathematics [ English ]

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1. Introduction

In set theory, the intersection operation is used to identify common elements between sets. While union combines elements, intersection focuses only on what is shared.

Intersection is widely used in:

• Mathematics
• Database filtering (AND conditions)
• Logical reasoning

2. Definition of Intersection

The intersection of two sets A and B is the set of all elements that belong to both A and B.

Formal Definition:

A ∩ B = { x | x ∈ A and x ∈ B }

3. Symbol and Meaning

• Symbol: ∩
• Read as: “A intersection B”
• Meaning: elements common to both sets

4. Example of Intersection

Let:

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

Then:

A ∩ B = {2, 3}

✔ Only common elements are included

5. Example with Disjoint Sets

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

A ∩ B = ∅

✔ No common elements → empty set

6. Properties of Intersection

(a) Commutative Law

A ∩ B = B ∩ A

(b) Associative Law

(A ∩ B) ∩ C = A ∩ (B ∩ C)

(c) Identity Law

A ∩ U = A

(d) Idempotent Law

A ∩ A = A

(e) Domination Law

A ∩ ∅ = ∅

7. Intersection with Complement

A ∩ A' = ∅

✔ A set and its complement have no common elements

8. Cardinality of Intersection

Intersection is used in union formula:

n(A ∩ B)

Example:

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

n(A ∩ B) = 2

9. Applications of Intersection

(a) Mathematics

• Finding common elements

(b) Computer Science

• Filtering data using AND conditions

(c) Real-Life

• Finding common members in groups

10. Key Observations

• Intersection includes only shared elements
• If no elements are common → result is empty set
• Works for all types of sets

11. Summary

• Intersection is written as A ∩ B

• Includes elements common to both sets

• If no common elements → ∅

• Follows laws:

  • Commutative
  • Associative
  • Identity

• Important relation:

  • A ∩ A' = ∅