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 \cup B = \{x : x \in A \text{ or } x \in 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 \cup B = \{1, 2, 3, 4, 5\}

— the 3 appears once.

Notation

A \cup 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 $A \cup B$ .

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 \cup B = \{x : x \in A \lor x \in B\}

Worked Examples

Example 1

easy

Let

A = \{1, 3, 5\}

and

B = \{2, 3, 4\}

. Find

A \cup B

Answer

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

First step

Recall the definition:

A \cup B = \{x : x \in A \text{ or } x \in B\}

. The word 'or' is inclusive — an element belongs to the union if it appears in at least one of the sets.

Full solution

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 $A$ ).
3
Therefore $A \cup B = \{1,2,3,4,5\}$ . Notice $|A \cup B| = 5 = |A| + |B| - |A \cap B| = 3 + 3 - 1$ , 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 \in \mathbb{R} : x > 2\}

and

B = \{x \in \mathbb{R} : x < 5\}

. Express

A \cup B

in interval notation.

Example 3

medium

A

and

B

are disjoint with

|A| = 5

and

|B| = 7

, find

|A \cup B|

Common Mistakes

Writing a shared element twice in the union — the union, being a set, lists each element once.
Confusing $\cup$ (or, combine) with $\cap$ (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 $|A \cap B|$ 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 $A \cup B$ .

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?

$A \cup B$

Why is the Union formula important in Math?

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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Related Concepts

Set Intersection Venn Diagram

Related Formulas

Set Formula Intersection Formula Venn Diagram Formula