Union Symbol (∪) and Intersection (∩): Meaning and LaTeX
July 17, 2026
The union symbol is ∪ — everything in either set. Its partner is intersection, ∩, meaning everything in both. A useful way to keep them straight: ∪ looks like a cup that holds everything, ∩ like a cap that covers only the overlap. Copy either below, or jump to how to type them.
| Character | Unicode | HTML entity | LaTeX | Windows alt code |
|---|---|---|---|---|
| ∪ union | U+222A | ∪ | \cup | — |
| ∩ intersection | U+2229 | ∩ | \cap | Alt + 239 |
What the symbol means
| Where | Meaning |
|---|---|
| Set theory | A ∪ B — elements in A, in B, or in both |
| Probability | P(A ∪ B) — the chance either event happens |
| Interval notation | Joining disjoint ranges, as in (−∞, 0) ∪ (0, ∞) |
| Databases | UNION combines the rows of two result sets |
| Topology | Arbitrary unions of open sets stay open |
Union and intersection
Union collects everything from both sets, keeping each element once. Intersection keeps only what appears in both. Both are commutative and associative, so the order and grouping never matter.
Counting a union: inclusion-exclusion
Adding the sizes of two sets double-counts the overlap, so you subtract it once. The same correction appears in probability, which is why the two formulas look identical.
The subtraction disappears only when the sets are disjoint — when A ∩ B is empty. That special case is where the common shortcut “just add the probabilities” comes from, and why it fails when the events can co-occur.
Interval notation and De Morgan
A union is how you write a domain with a hole in it — the two pieces cannot be a single interval, so they are joined. De Morgan’s laws then say that negating a union swaps it for an intersection of the complements.
How to type it
Windows
- ∩: hold
Altand type239on the numeric keypad - ∪ has no legacy alt code. In Word, type
222Aand pressAlt + X - Elsewhere, press
Windows + .and search “union”
Mac
- Open Character Viewer with
Control + Command + Spaceand search “union” or “intersection”
HTML
∪or∪for ∪∩or∩for ∩
In LaTeX
\cup and \cap are binary operators. \bigcup and \bigcap are the large operator forms that take limits, for a union over an index set.
Do not use a capital U for union. It is an italic variable, not an operator, so it gets the wrong shape and no operator spacing — an immediate tell in a submitted paper.
Frequently asked questions
What does ∪ mean?
Union — the set of everything that appears in either set, with duplicates counted once.
What is the difference between ∪ and ∩?
∪ is union: in either set. ∩ is intersection: in both. The cup holds everything; the cap covers only the overlap.
How do I write a union of intervals?
Join them with ∪, as in (-∞, 0) ∪ (0, ∞). This is the standard way to write a domain that excludes a point.
See also the real numbers symbol, the not equal symbol and mathematical symbols in LaTeX.
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