Union Formula

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

The Formula

A∪B={x:x∈A or x∈B}

When to use: Pour both sets into one container and remove duplicates. Everything from either pile ends up in the union — this is the OR operation for sets.

Quick Example

A={1,2,3}, B={3,4,5}. Then A∪B={1,2,3,4,5} — the 3 appears once.

Notation

A∪B

What This Formula Means

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

Pour both sets into one container and remove duplicates. Everything from either pile ends up in the union — this is the OR operation for sets.

Formal View

A∪B={x:x∈A∨x∈B}

Worked Examples

Example 1

easy
Let A={1,3,5} and B={2,3,4}. Find A∪B.

Answer

A∪B={1,2,3,4,5}

First step

1
Recall the definition: A∪B={x:x∈A or x∈B}. The word 'or' is inclusive — an element belongs to the union if it appears in at least one of the sets.

Full solution

  1. 2
    List all elements from A={1,3,5} and B={2,3,4}, including each at most once: 1 (from A), 2 (from B), 3 (in both), 4 (from B), 5 (from A).
  2. 3
    Therefore A∪B={1,2,3,4,5}. Notice ∣A∪B∣=5=∣A∣+∣B∣−∣A∩B∣=3+3−1, confirming the inclusion-exclusion principle.
The union operator collects all elements from both sets. Duplicate elements are listed only once because sets contain distinct elements.

Example 2

medium
Let A={x∈R:x>2} and B={x∈R:x<5}. Express A∪B in interval notation.

Example 3

medium
If A and B are disjoint with ∣A∣=5 and ∣B∣=7, find ∣A∪B∣.

Common Mistakes

  • Writing a shared element twice in the union — the union, being a set, lists each element once.
  • Confusing ∪ (or, combine) with ∩ (and, overlap) — union grows or stays the same, intersection shrinks or stays the same.
  • Sizing a union as ∣A∣+∣B∣ when the sets overlap — subtract ∣A∩B∣ to avoid double-counting.

Why This Formula Matters

Union is the OR of set theory and feeds straight into probability ('A or B happens'), counting with inclusion-exclusion, and database queries. A student who double-counts the overlap when listing or sizing a union will overstate every combined count. Recognizing it by "Does an item belong as long as it is in at least one of the sets?" — rather than by familiar numbers — is what lets a student tell it apart from intersection and sum of cardinalities and concatenation of lists in a mixed problem set.

Frequently Asked Questions

What is the Union formula?

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

How do you use the Union formula?

Pour both sets into one container and remove duplicates. Everything from either pile ends up in the union — this is the OR operation for sets.

What do the symbols mean in the Union formula?

A∪B

Why is the Union formula important in Math?

Union is the OR of set theory and feeds straight into probability ('A or B happens'), counting with inclusion-exclusion, and database queries. A student who double-counts the overlap when listing or sizing a union will overstate every combined count. Recognizing it by "Does an item belong as long as it is in at least one of the sets?" — rather than by familiar numbers — is what lets a student tell it apart from intersection and sum of cardinalities and concatenation of lists in a mixed problem set.

What do students get wrong about Union?

The procedure for union is the easy part; the trap is writing a shared element twice in the union. Asking "Does an item belong as long as it is in at least one of the sets?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Union formula?

Before studying the Union formula, you should understand: set.

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