Intersection in LaTeX

The intersection symbol ∩ is \cap in math mode: $A \cap B$ . To intersect an indexed family of sets, use the big operator \bigcap with limits: \bigcap_{i=1}^{n} A_i.

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Intersection commands

\cap
$\cap$
Intersection of two sets (binary operator).
\bigcap
$⋂$
Big intersection over an index.
\bigcap_{i=1}^{n}
$⋂_{i = 1}^{n}$
Intersection for i from 1 to n.
\bigcap_{k \in \mathbb{N}}
$⋂_{k \in N}$
Intersection over an index set.
\sqcap
$⊓$
Square cap, used for meets in lattices.
\Cap
$⋒$
Double intersection (amssymb).

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LaTeX

Templates

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Intersection examples

Each result below is drawn from the LaTeX under it. Copy the code, edit it on this page, or open it in the full editor.

Two sets
$A \cap B = {x : x \in A and x \in B}$
LaTeX
```
A \cap B = \{ x : x \in A \text{ and } x \in B \}
```
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Indexed intersection
$⋂_{i = 1}^{n} A_{i}$
LaTeX
```
\bigcap_{i=1}^{n} A_i
```
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Nested intervals
$⋂_{n = 1}^{\infty} (0, \frac{1}{n}) = \emptyset$
LaTeX
```
\bigcap_{n=1}^{\infty} \left( 0, \frac{1}{n} \right) = \varnothing
```
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Disjoint sets
$A \cap B = \emptyset$
LaTeX
```
A \cap B = \varnothing
```
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De Morgan’s law
$\overset{―}{A \cap B} = \overset{―}{A} \cup \overset{―}{B}$
LaTeX
```
\overline{A \cap B} = \overline{A} \cup \overline{B}
```
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Probability of both events
$P (A \cap B) = P (A) P (B ∣ A)$
LaTeX
```
P(A \cap B) = P(A)\,P(B \mid A)
```
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Small ∩ or big ⋂

Use \cap between two sets. For "the intersection of all A_i", use \bigcap and put the index in the limits, just like \sum. The big operator scales with display style: large with limits under and over it in \[ … \], smaller with side limits inline.

If \cap looks too small next to large sets in a displayed formula, that is by design: it is a binary operator. Switch to \bigcap only when there is an index; for two big expressions keep \cap and let spacing do the work.

Intersections in probability

Probability texts write joint events as P(A \cap B). Conditional probability uses \mid for the bar, P(B \mid A), which gets proper spacing, unlike a plain |.

Common mistakes

A lowercase n is a variable letter and is spaced like one.
Instead of
```
A n B
```
Write
```
A \cap B
```
Scripts on the binary \cap sit beside a small symbol; the big operator stacks them properly.
Instead of
```
\cap_{i=1}^{n} A_i
```
Write
```
\bigcap_{i=1}^{n} A_i
```
LaTeX names the symbol by its shape, cap. \intersect and \intersection do not exist.
Instead of
```
\intersect
```
Write
```
\cap
```

In Overleaf, MathJax and Word

Overleaf and pdfLaTeX
\cap, \bigcap and \sqcap are built in; \Cap needs amssymb.
MathJax and KaTeX
MathJax and KaTeX render all of them; \bigcap limits follow display and inline style as in LaTeX.
Microsoft Word
In a Word equation, \cap then Space inserts ∩. For an indexed intersection, build it here and use Copy for Word to paste an editable equation.