Intersection (set theory)

In set theory, the intersection of two sets $A$ and $B,$ denoted by $A \cap B,$ 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 symbol "$\cap$" between the terms; that is, in infix notation. For example:
$\\cap\=\$
$\\cap\=\varnothing$
$\Z\cap\N=\N$
$\\cap\N=\$
The intersection of more than two sets (generalized intersection) can be written as:
$\bigcap_^n A_i$
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 $A \cap B$, is the set of all objects that are members of both the sets $A$ and $B.$
In symbols:
$A \cap B = \.$

That is, $x$ is an element of the intersection $A \cap B$ if and only if $x$ is both an element of $A$ and an element of $B.$

For example:
* The intersection of the sets and is .
* The number 9 is in the intersection of the set of prime numbers and the set of odd numbers , because 9 is not prime.

Intersecting and disjoint sets

We say that if there exists some $x$ that is an element of both $A$ and $B,$ in which case we also say that . Equivalently, $A$ intersects $B$ if their intersection $A \cap B$ is an , meaning that there exists some $x$ such that $x \in A \cap B.$

We say that 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 $\$ and $\$ are disjoint, while the set of even numbers intersects the set of multiples of 3 at the multiples of 6.

Algebraic properties

Binary intersection is an associative operation; that is, for any sets $A, B,$ and $C,$ one has

$A \cap (B \cap C) = (A \cap B) \cap C.$Thus the parentheses may be omitted without ambiguity: either of the above can be written as $A \cap B \cap C$. Intersection is also commutative. That is, for any $A$ and $B,$ one has$A \cap B = B \cap A.$
The intersection of any set with the empty set results in the empty set; that is, that for any set $A$,
$A \cap \varnothing = \varnothing$
Also, the intersection operation is idempotent; that is, any set $A$ satisfies that $A \cap A = A$. All these properties follow from analogous facts about logical conjunction.

Intersection distributes over union and union distributes over intersection. That is, for any sets $A, B,$ and $C,$ one has
$\begin A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \\ A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \end$
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:$A \cap B = \left(A^ \cup B^\right)^c$

Arbitrary intersections

The most general notion is the intersection of an arbitrary collection of sets.
If $M$ is a nonempty set whose elements are themselves sets, then $x$ is an element of the of $M$ if and only if for every element $A$ of $M,$ $x$ is an element of $A.$
In symbols:
$\left( x \in \bigcap_ A \right) \Leftrightarrow \left( \forall A \in M, \ x \in A \right).$

The notation for this last concept can vary considerably. Set theorists will sometimes write "$\cap M$", while others will instead write "$\cap_ A$".
The latter notation can be generalized to "$\cap_ A_i$", which refers to the intersection of the collection $\left\.$
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:
$\bigcap_^ A_i.$

When formatting is difficult, this can also be written "$A_1 \cap A_2 \cap A_3 \cap \cdots$". 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 ($\varnothing$). The reason is as follows: The intersection of the collection $M$ is defined as the set (see set-builder notation)
$\bigcap_ A = \.$
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 . 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 ( ZF) set theory, the universal set does not exist.

In type theory however, $x$ is of a prescribed type $\tau,$ so the intersection is understood to be of type $\mathrm\ \tau$ (the type of sets whose elements are in $\tau$), and we can define $\bigcap_ A$ to be the universal set of $\mathrm\ \tau$ (the set whose elements are exactly all terms of type $\tau$).

See also

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References

Further reading

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External links

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{{Mathematical logic

Basic concepts in set theory
Operations on sets
Intersection