# Intersection (set theory)

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.

## Basic definition

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

that is

*x*∈*A*∩*B*if and only if*x*∈*A*and*x*∈*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, …}.^{[2]}

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 complement *A*^{c} of *A* to be the set of all elements of *U* not in *A*. Now 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:

*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 .

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.

## 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 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 (⋂).

## 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) ^{[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.

## See also

- Complement
- Intersection graph
- Logical conjunction
- Naive set theory
- Symmetric difference
- Union
- Cardinality
- Iterated binary operation
- MinHash

## References

- ↑ Template:Cite web
- ↑ How to find the intersection of sets
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}

## Further reading

- Devlin, K.J.,
*The Joy of Sets: Fundamentals of Contemporary Set Theory*, 2nd edition, Springer-Verlag, New York, NY, 1993. - "Chapter 1" Munkres, James R. Topology. 2nd edition. Upper Saddle River: Prentice Hall, 2000.
- {{#invoke:citation/CS1|citation

|CitationClass=book }}