set-theoretic intersection

In set theory, the intersection of two Set (mathematics), 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 set, empty, denoted $A \cap B = \varnothing.$

For example, the sets $\$ and $\$ are disjoint, while the set of even numbers intersects the set of Multiple (mathematics), 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 property, 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 Idempotence, idempotent; that is, any set $A$ satisfies that $A \cap A = A$. All these properties follow from analogous facts about logical conjunction.

Intersection Distributive property, distributes over Union (set theory), 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 (set theory), 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 (set theory), 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 Empty set, nonempty set whose elements are themselves sets, then $x$ is an element of the of $M$ if and only if Universal quantification, 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 theory, 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 Sigma algebra, 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 (Zermelo–Fraenkel set theory, 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