Cup (Union Symbol) — Definition, Formula & Examples

The cup symbol (∪) stands for the union of two sets, meaning you combine all elements from both sets into one set, without repeating any element.

For sets $A$ and $B$ , the union $A \cup B$ is the set of all elements $x$ such that $x \in A$ or $x \in B$ (or both). In set-builder notation: $A \cup B = \{x \mid x \in A \text{ or } x \in B\}$ .

Key Formula

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

Where:

$A$ = The first set
$B$ = The second set
$x$ = Any element that belongs to at least one of the two sets

How It Works

To find

A \cup B

, list every element that appears in

A

, in

B

, or in both. If the same element appears in both sets, you write it only once in the result. The cup symbol (∪) opens upward like a cup collecting items — this visual can help you remember it means union.

Worked Example

Problem: Let A = {1, 2, 3} and B = {2, 4, 6}. Find A ∪ B.

Step 1: List all elements from set A.

1, 2, 3

Step 2: Add any elements from set B that are not already listed. The element 2 is already included, so add only 4 and 6.

1, 2, 3, 4, 6

Step 3: Write the result as a set.

A \cup B = \{1, 2, 3, 4, 6\}

Answer:

A \cup B = \{1, 2, 3, 4, 6\}

Why It Matters

Union comes up whenever you need to combine groups — for example, merging two class rosters or combining search results from two databases. Understanding the ∪ symbol is essential for Venn diagram problems in middle school and for probability, where

P(A \cup B)

describes the chance that at least one of two events occurs.

Common Mistakes

Mistake: Confusing the cup (∪) for union with the cap (∩) for intersection.

Correction: Remember that the cup opens upward like a container collecting everything. The cap (∩) points downward and keeps only what overlaps. Union = all elements; intersection = shared elements only.