Set Union

Calculator for computing set unions with comprehensive formulas and examples

Set Union Calculator

What is calculated?

The union A ∪ B contains all elements that are in at least one of the two sets A or B. It forms the "total" of both sets without duplicates.

Enter the Sets

Set A
Values separated by spaces or semicolons

Set B
Values separated by spaces or semicolons

Result (A ∪ B)

Union:

All elements from both sets (without duplicates)

Set Union Info

Properties

Set Union A ∪ B:

Commutative: A ∪ B = B ∪ A
Associative: (A ∪ B) ∪ C = A ∪ (B ∪ C)
Idempotent: A ∪ A = A
A ⊆ (A ∪ B) and B ⊆ (A ∪ B)

Venn diagram of set union

Memory aid: "All elements that are in at least one of the two sets."

Inclusion-Exclusion Principle

|A ∪ B| = |A| + |B| - |A ∩ B|

Number of elements in the union

Examples

Standard:
{1,2,3} ∪ {2,3,4} = {1,2,3,4}

Disjoint:
{1,2} ∪ {3,4} = {1,2,3,4}

Subset:
{1,2} ∪ {1,2,3} = {1,2,3}

Related Operations

→ Set Intersection
→ Set Difference
→ Symmetric Difference

Set Union Formulas

Basic Definition

\[A \cup B = \{x : x \in A \lor x \in B\}\] Set notation

Commutative Law

\[A \cup B = B \cup A\] Order doesn't matter

Associative Law

\[(A \cup B) \cup C = A \cup (B \cup C)\] Grouping doesn't matter

Distributive Laws

\[A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\] Distribution over intersection

Idempotence

\[A \cup A = A\] Self union

Neutral/Absorbing Elements

\[A \cup \emptyset = A\] \[A \cup U = U\] With empty set and universal set

Detailed Calculation Example

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

Given:

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

Step 1 - Collect all elements:

From A: 1, 2, 3, 4, 5, 6

From B: 4, 5, 6, 7, 8, 9

Total: 1, 2, 3, 4, 5, 6, 7, 8, 9

Step 2 - Remove duplicates:

A ∪ B = {1,2,3,4,5,6,7,8,9}

Verification with cardinality:

|A| = 6, |B| = 6

|A ∩ B| = |{4,5,6}| = 3

|A ∪ B| = 6 + 6 - 3 = 9 ✓

Interpretation: The union contains all distinct elements from both sets, with common elements counted only once.

Inclusion-Exclusion Principle

The fundamental principle for calculating cardinalities

For two sets:

|A ∪ B| = |A| + |B| - |A ∩ B|

Reason: Common elements would otherwise be counted twice

For three sets:

|A ∪ B ∪ C| = |A| + |B| + |C|
- |A ∩ B| - |A ∩ C| - |B ∩ C|
+ |A ∩ B ∩ C|

Practical Example

Survey in a class of 30 students:

Sports:
Soccer: 18 students
Basketball: 12 students
Both sports: 8 students

Calculation:
At least one sport:
|S ∪ B| = 18 + 12 - 8 = 22 students

Result: 22 out of 30 students play at least one of the two sports.

Set Union Laws

Fundamental laws of set theory

Absorption Law

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

Absorption of intersection

De Morgan 1

\[\overline{A \cup B} = \overline{A} \cap \overline{B}\]

Complement of union

Distributive Law

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

Distribution over intersection

Monotonicity

\[\text{If } A \subseteq B, \text{ then } A \cup C \subseteq B \cup C\]

Monotonicity property

Example for Absorption Law

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

A ∩ B = {2,3}
A ∪ (A ∩ B) = {1,2,3} ∪ {2,3} = {1,2,3} = A ✓

Mathematical Properties

Basic Properties

Upper bound: A ⊆ (A ∪ B) and B ⊆ (A ∪ B)
Commutativity: A ∪ B = B ∪ A
Associativity: (A ∪ B) ∪ C = A ∪ (B ∪ C)
Idempotence: A ∪ A = A

Special Properties

Neutral element: A ∪ ∅ = A
Absorbing element: A ∪ U = U
Complement: A ∪ A^c = U
Least upper bound: A ∪ B is the smallest set containing A and B

Important Notes

Size relationship: |A ∪ B| ≥ max(|A|, |B|)

Supremum: A ∪ B is the supremum (least upper bound) of A and B

Practical Applications

Databases

SQL UNION operations
Merging datasets
Aggregated queries
Master data management

Social Networks

Combining friend lists
Interest groups
Recommendation systems
Community building

Data Analysis

Creating total populations
Combining categories
Survey evaluations
Market research

Set theory functions

Complement • Difference • Intersection • Symmetric Difference • Union •