Union of Sets in Discrete Mathematics [ English ]

Languages

• English

1. Introduction

In set theory, one of the most fundamental operations is the union of sets. It is used to combine elements from multiple sets into a single set.

The union operation is widely applied in:

• Mathematics
• Database queries
• Logic and programming

2. Definition of Union

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

Formal Definition:

A ∪ B = { x | x ∈ A or x ∈ B }

3. Symbol and Meaning

• Symbol: ∪
• Read as: “A union B”
• Meaning: combine elements of both sets without repetition

4. Example of Union

Let:

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

Then:

A ∪ B = {1, 2, 3, 4, 5}

✔ Note:

• Element 3 appears only once
• Sets do not allow duplicates

5. Example with Disjoint Sets

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

A ∪ B = {1, 2, 3, 4}

✔ Since sets are disjoint, all elements are simply combined

6. Properties of Union

(a) Commutative Law

A ∪ B = B ∪ A

(b) Associative Law

(A ∪ B) ∪ C = A ∪ (B ∪ C)

(c) Identity Law

A ∪ ∅ = A

(d) Idempotent Law

A ∪ A = A

(e) Domination Law

A ∪ U = U

Where U is the universal set.

7. Union with Complement

A ∪ A' = U

✔ Union of a set and its complement gives the universal set

8. Cardinality of Union

Number of elements in union:

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Example

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

n(A ∪ B) = 3 + 2 − 1 = 4

9. Applications of Union

(a) Mathematics

• Combining sets

(b) Computer Science

• Query results (OR condition)

(c) Real-Life

• Combining groups or categories

10. Key Observations

• Union includes all elements from both sets
• Duplicate elements are counted only once
• Works for both finite and infinite sets

11. Summary

• Union is written as A ∪ B

• Includes elements in A, B, or both

• Follows important laws:

  • Commutative
  • Associative
  • Identity

• Cardinality formula:

  • n(A ∪ B) = n(A) + n(B) − n(A ∩ B)