Intersection (∩) in LaTeX
\cap typesets the intersection symbol ∩; is the set of elements that belong to both A and B.
LaTeX command
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Intersection collects only what two sets share. keeps an element x precisely when x is in A and x is in B; anything in only one of the sets is dropped. Like union, intersection is commutative and associative, so a chain needs no parentheses.
When two sets share nothing at all, their intersection is the empty set, written = \emptyset; this is the standard way to state that A and B are disjoint. Intersection also has an identity element with the universal set U, since = A.
For intersecting a whole family of sets, use \bigcap instead of writing out \cap repeatedly. \bigcap_{i=1}^n A_i and \bigcap_{i \in I} A_i both take limits exactly like \bigcup and \sum, making chains of intersections easy to read even with many terms.
Other forms
\bigcapbig intersection (⋂)Examples
$A \cap B$everything in both A and B
$A \cap B = \emptyset$A and B are disjoint
$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$the distributive law
$\bigcap_{i=1}^{n} A_i$an intersection of many sets, with limits like \sum
$A \cap A = A$intersection is idempotent
Tips and common mistakes
- Use
\capfor two (or a short explicit list of) sets and\bigcapfor an intersection over an index, mirroring the\cupand\bigcuppattern. - =
\emptysetis the standard way to state that two sets are disjoint; check for it before assuming an element belongs to both sets in a proof. \bigcaptakes the same subscript/superscript syntax as\sumand\bigcup:\bigcap_{i \in I}A_i.- Do not confuse
\cap(set intersection) with\wedge(logical and); use\capfor sets and\wedgeor\landfor propositions. - Remember the distributive laws linking
\capand\cup, A\cap() = ()\cup() and its mirror, they are the most commonly needed identity in set proofs.
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What does mean?
It is the intersection of A and B: the set of elements that belong to both A and B at once. Anything in only one of the two sets is excluded.
How do I show two sets are disjoint in LaTeX?
Write = \emptyset, which states that the two sets share no elements.
What is the difference between \cap and \bigcap?
\cap intersects two explicitly named sets, like . \bigcap is the big-operator form for intersecting an indexed family of sets, written \bigcap_{i \in I} A_i, taking limits the same way \sum does.
Is \cap the same as logical and?
No, though they behave similarly. \cap combines sets; \wedge (or \land) connects logical propositions. Keep them separate, they are not interchangeable typographically or conceptually.