Set Intersection

Calculator for computing set intersections with comprehensive formulas and examples

Set Intersection Calculator

What is calculated?

The intersection A ∩ B contains all elements that are in both set A and set B. It represents the "common" elements of both sets.

Enter the Sets

Values separated by spaces or semicolons

Values separated by spaces or semicolons
Result (A ∩ B)
Intersection:
Elements that appear in both sets

Set Intersection Info

Properties

Set Intersection A ∩ B:

  • Commutative: A ∩ B = B ∩ A
  • Associative: (A ∩ B) ∩ C = A ∩ (B ∩ C)
  • Idempotent: A ∩ A = A
  • A ∩ B ⊆ A and A ∩ B ⊆ B
Set Intersection Venn Diagram

Venn diagram of set intersection

Memory aid: "The intersection contains only elements that appear in both sets simultaneously."

Examples
Standard:
{1,2,3} ∩ {2,3,4} = {2,3}
Disjoint:
{1,2} ∩ {3,4} = ∅
Subset:
{1,2} ∩ {1,2,3} = {1,2}
Related Operations

→ Set Union
→ Set Difference
→ Symmetric Difference


Set Intersection Formulas

Basic Definition
\[A \cap B = \{x : x \in A \land x \in B\}\] Set notation
Commutative Law
\[A \cap B = B \cap A\] Order doesn't matter
Associative Law
\[(A \cap B) \cap C = A \cap (B \cap C)\] Grouping doesn't matter
Distributive Laws
\[A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\] Distribution over union
Idempotence
\[A \cap A = A\] Self intersection
Neutral Elements
\[A \cap U = A\] \[A \cap \emptyset = \emptyset\] With universal and empty set

Detailed Calculation Example

Example: A = {1,2,3,4,5}, B = {4,5,6,7,8,9}

Given:

  • A = {1, 2, 3, 4, 5}
  • B = {4, 5, 6, 7, 8, 9}

Step 1 - Check elements:

1 ∈ A, but 1 ∉ B ✗
2 ∈ A, but 2 ∉ B ✗
3 ∈ A, but 3 ∉ B ✗
4 ∈ A and 4 ∈ B ✓
5 ∈ A and 5 ∈ B ✓

Step 2 - Common elements:

A ∩ B = {4, 5}

Verification:

B ∩ A = {4, 5} ✓ (commutative)
|A ∩ B| = 2 elements

Interpretation: The intersection contains only elements 4 and 5, as these are the only ones that appear in both sets.

Practical Application Example

Example: Shared hobbies of friends

Anna's hobbies (A):

Reading, Swimming, Cycling, Cooking, Painting

Ben's hobbies (B):

Swimming, Hiking, Cooking, Gaming, Photography

Question: What hobbies do Anna and Ben have in common?

Answer (A ∩ B): {Swimming, Cooking}

Result: Anna and Ben have 2 shared hobbies where they can do activities together.

Set Intersection Laws

Fundamental laws of set theory
Commutative Law
\[A \cap B = B \cap A\]

Order doesn't matter

Associative Law
\[(A \cap B) \cap C = A \cap (B \cap C)\]

Grouping doesn't matter

Distributive Law
\[A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\]

Distribution over union

Absorption Law
\[A \cap (A \cup B) = A\]

Absorption of union

Example for Distributive Law

Given: A = {1,2}, B = {2,3}, C = {3,4}

Left side:
B ∪ C = {2,3,4}
A ∩ (B ∪ C) = {1,2} ∩ {2,3,4} = {2}
Right side:
A ∩ B = {2}, A ∩ C = ∅
(A ∩ B) ∪ (A ∩ C) = {2} ∪ ∅ = {2} ✓

Mathematical Properties

Basic Properties
  • Subset: A ∩ B ⊆ A and A ∩ B ⊆ B
  • Commutativity: A ∩ B = B ∩ A
  • Associativity: (A ∩ B) ∩ C = A ∩ (B ∩ C)
  • Idempotence: A ∩ A = A
Special Properties
  • Empty set: A ∩ ∅ = ∅
  • Universal set: A ∩ U = A
  • Complement: A ∩ Ac = ∅
  • Cardinality: |A ∩ B| ≤ min(|A|, |B|)
Important Notes

Disjoint sets: If A ∩ B = ∅, then A and B are disjoint

Greatest lower bound: A ∩ B is the largest set contained in both

Practical Applications

Databases
  • SQL INTERSECT operations
  • JOIN operations
  • Finding common records
  • Overlap analysis
Social Analysis
  • Mutual friends
  • Overlapping interests
  • Shared attributes
  • Group intersections
Data Analysis
  • Overlapping categories
  • Common features
  • Filter operations
  • Correlation analysis




Set theory functions

Complement  •  Difference  •  Intersection  •  Symmetric Difference  •  Union  •