Set Union
The union of two sets A and B is the set containing all elements that are in A, in B, or in both, with no element repeated. Written as A ∪ B, it combines the members of both sets into a single collection. Set union is used in probability (combining events), database queries (SQL OR), and logic circuit design.
Key Formula
A ∪ B = { x | x is in A or x is in B }
LaTeX: A \cup B = \{ x \mid x \in A \text{ or } x \in B \}
| Symbol | Meaning | Unit |
|---|---|---|
| A | First set | set |
| B | Second set | set |
| x | An arbitrary element | element |
Worked Example
Problem
Let A = {2, 4, 6, 8} and B = {1, 2, 3, 4}. Find A ∪ B and |A ∪ B| using the inclusion-exclusion principle.
Solution
Step 1: List all distinct elements from both sets. — A ∪ B = {1, 2, 3, 4, 6, 8} Step 2: Use the inclusion-exclusion principle: |A ∪ B| = |A| + |B| − |A ∩ B| — |A| = 4, |B| = 4 — A ∩ B = {2, 4}, so |A ∩ B| = 2 — |A ∪ B| = 4 + 4 − 2 = 6
Answer
A ∪ B = {1, 2, 3, 4, 6, 8}, with |A ∪ B| = 6 elements.
Properties of Set Union
| Property | Formula | Description |
|---|---|---|
| Commutativity | A ∪ B = B ∪ A | Order of sets does not matter |
| Associativity | (A ∪ B) ∪ C = A ∪ (B ∪ C) | Grouping does not affect result |
| Identity | A ∪ ∅ = A | Union with empty set returns original set |
| Idempotency | A ∪ A = A | Union of a set with itself is the set |
| Domination | A ∪ U = U | Union with universal set is universal set |
| Inclusion-Exclusion | |A ∪ B| = |A| + |B| − |A ∩ B| | Cardinality formula |
Interactive Tools
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Related Terms
Set Theory
Set theory is the branch of mathematical logic that studies collections of objects, called sets, and the relationships between them. It provides the foundational language for nearly all of modern mathematics, defining concepts like numbers, functions, and relations in terms of sets. Developed formally by Georg Cantor in the 1870s, it underpins areas from algebra and topology to computer science and logic.
Set Intersection
The intersection of two sets A and B is the set containing only those elements that are members of both A and B simultaneously. Written as A ∩ B, it identifies common elements shared between collections. Set intersection is used in database queries (SQL JOIN/AND), probability (joint events), and data deduplication tasks.
Subset
A subset is a set whose every element is also an element of another set, called the superset. If A is a subset of B (written A ⊆ B), then for every element x in A, x is also in B. The concept of subsets is fundamental to set theory and appears throughout mathematics in topics like topology, algebra, and probability theory.
From Latin "unio" (oneness, unity), itself from "unus" (one). In mathematics, the union symbol ∪ was introduced by Giuseppe Peano in his 1888 work "Calcolo geometrico".