# Intersection (set theory)

#### Background Information

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In mathematics, the **intersection** 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.

*For explanation of the symbols used in this article, refer to the table of mathematical symbols.*

## Basic definition

The intersection of *A* and *B* is written "*A* ∩ *B*". Formally:

*x*is an element of*A*∩*B*if and only if*x*is an element of*A*and*x*is an element of*B*.

- For example:
- The intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}.
- The number 9 is
*not*in the intersection of the set of prime numbers {2, 3, 5, 7, 11, …} and the set of odd numbers {1, 3, 5, 7, 9, 11, …}.

If the intersection of two sets *A* and *B* is empty, that is they have no elements in common, then they are said to be **disjoint**, denoted: *A* ∩ *B* = Ø. For example the sets {1, 2} and {3, 4} are disjoint, written

{1, 2} ∩ {3, 4} = Ø.

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*.

## 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:

This idea subsumes the above paragraphs, in that for example, *A* ∩ *B* ∩ *C* is the intersection of the collection {*A*,*B*,*C*}.

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, you might see notation analogous to that of an infinite series:

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 (⋂).

## 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 "set of everything". The problem is, *there is no such set*. Assuming such a set exists leads to a famous problem in naive set theory known as Russell's paradox. For this reason the intersection of the empty set is left undefined.

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.