Intersection (set theory)

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In
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
, 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 Infix notation is the notation commonly used in arithmetical and logical formulae and statements. It is characterized by the placement of operators between operands—" infixed operators"—such as the plus sign in . Usage Binary relations are ...
. 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 In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, matr ...
. For an explanation of the symbols used in this article, refer to the
table of mathematical symbols A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula ...
.

# 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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
$x$ 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 number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s and the set of
odd numbers In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 ...
, 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 In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacemen ...
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 In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
. 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 In logic, mathematics and linguistics, And (\wedge) is the truth-functional operator of logical conjunction; the ''and'' of a set of operands is true if and only if ''all'' of its operands are true. The logical connective that represents this ...
. 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 In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
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 number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
s, notation analogous to that of an
infinite product In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product : \prod_^ a_n = a_1 a_2 a_3 \cdots is defined to be the limit of the partial products ''a''1''a''2...''a'n'' as ''n'' increases without bound. ...
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 In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy. Definin ...
) $\bigcap_ A = \.$ 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 In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. For example, the statement "sh ...
. So the intersection of the empty family should be the
universal set In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory inc ...
(the identity element for the operation of intersection), but in standard ( ZF) set theory, the universal set does not exist. In
type theory In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a founda ...
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 $\tau$).