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In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.^{[1]}
For explanation of the symbols used in this article, refer to the table of mathematical symbols.
The intersection of A and B is written "A ∩ B". Formally:
that is
For example:
More generally, one can take the intersection of several sets at once. The intersection of A, B, C, and D, for example, is A ∩ B ∩ C ∩ D = A ∩ (B ∩ (C ∩ D)). Intersection is an associative operation; thus, A ∩ (B ∩ C) = (A ∩ B) ∩ C.
Inside a universe U one may define the 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 A\cap B=\varnothing.
For example, the sets {1, 2} and {3, 4} are disjoint, the set of even numbers intersects the set of multiples of 3 at 0, 6, 12, 18 and other numbers.
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 series may be seen:
When formatting is difficult, this can also be written "A_{1} ∩ A_{2} ∩ A_{3} ∩ ...", even though strictly speaking, A_{1} ∩ (A_{2} ∩ (A_{3} ∩ ... makes no sense. (This last example, an intersection of countably many sets, is actually very common; for an example see the article on σ-algebras.)
Finally, let us note that whenever the symbol "∩" is placed before other symbols instead of between them, it should be of a larger size (⋂).
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) ^{[3]}
Unfortunately, according to standard (ZFC) set theory, the universal set does not exist. A partial fix for this problem can be found if we agree to restrict our attention to subsets of a fixed set U called the universe. In this case the intersection of a family of subsets of U can be defined as
Now if M is empty there is no problem. The intersection is just the entire universe U, which is a well-defined set by assumption and becomes the identity element for this operation.
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