Intersection (set theory)

In mathematics, the intersection of two sets "A" and "B", denoted by "A" ∩ "B", is the set containing all elements of "A" that also belong to "B" (or equivalently, all elements of "B" that also belong to "A").
Notation and terminology.
Intersection is written using the sign "∩" between the terms; that is, in infix notation. For example,
The intersection of more than two sets (generalized intersection) can be written as
which is similar to capital-sigma notation. 
For an explanation of the symbols used in this article, refer to the table of mathematical symbols.
Definition.
The intersection of two sets "A" and "B", denoted by , is the set of all objects that are members of both the sets and .
In symbols,
That is, "x" is an element of the intersection "A" ∩ "B", if and only if "x" is both an element of "A" and an element of "B".
For example:
Intersection is an associative operation; that is, for any sets "A", "B", and "C", one has "A" ∩ ("B" ∩ "C") = ("A" ∩ "B") ∩ "C". Intersection is also commutative; for any "A" and "B", one has "A" ∩ "B" = "B" ∩ "A." It thus makes sense to talk about intersections of multiple sets. The intersection of "A", "B", "C", and "D", for example, is unambiguously written "A" ∩ "B" ∩ "C" ∩ "D".
Inside a universe "U", one may define the complement "A"c of "A" to be the set of all elements of "U" not in "A". Furthermore, the intersection of "A" and "B" may be written as the complement of the union of their complements, derived easily from De Morgan's laws:<br>
"A" ∩ "B" = ("A"c ∪ "B"c)c
Intersecting and disjoint sets.
We say that "A intersects (meets) B at an element x" if "x" belongs to "A" and "B". We say that "A intersects (meets) B" if "A" intersects B at some element. "A" intersects "B" if their intersection is inhabited.
We say that "A and B are disjoint" if "A" does not intersect "B". In plain language, they have no elements in common. "A" and "B" are disjoint if their intersection is empty, denoted formula_7.
For example, the sets {1, 2} and {3, 4} are disjoint, while the set of even numbers intersects the set of multiples of 3 at the multiples of 6.
Arbitrary intersections.
The most general notion is the intersection of an arbitrary "nonempty" collection of sets.
If "M" is a nonempty set whose elements are themselves sets, then "x" is an element of the "intersection" of "M" if and only if for every element "A" of "M", "x" is an element of "A".
In symbols:
The notation for this last concept can vary considerably. Set theorists will sometimes write "⋂"M"", while others will instead write "⋂"A"∈"M" "A"".
The latter notation can be generalized to "⋂"i"∈"I" "A""i"", which refers to the intersection of the collection {"A""i" : "i" ∈ "I"}.
Here "I" is a nonempty set, and "A""i" is a set for every "i" in "I".
In the case that the index set "I" is the set of natural numbers, notation analogous to that of an infinite product may be seen:
When formatting is difficult, this can also be written ""A"1 ∩ "A"2 ∩ "A"3 ∩ ...". This last example, an intersection of countably many sets, is actually very common; for an example, see the article on σ-algebras.
Nullary intersection.
Note that in the previous section, we excluded the case where "M" was the empty set (∅). The reason is as follows: The intersection of the collection "M" is defined as the set (see set-builder notation)
If "M" is empty, there are no sets "A" in "M", so the question becomes "which "x"'s satisfy the stated condition?" The answer seems to be "every possible x". When "M" is empty, the condition given above is an example of a vacuous truth. So the intersection of the empty family should be the universal set (the identity element for the operation of intersection),
but in standard (ZFC) set theory, the universal set does not exist.
In type theory however, "x" is of a prescribed type formula_11, so the intersection is understood to be of type formula_12 (the type of sets whose elements are in formula_11), and we can define formula_14 to be the universal set of formula_12 (the set whose elements are exactly all terms of type formula_11).