Intersection (set theory)

In set theory, the intersection of two sets ${\displaystyle A}$ and ${\displaystyle B,}$ denoted by ${\displaystyle A\cap B,}$^[1] is the set containing all elements of ${\displaystyle A}$ that also belong to ${\displaystyle B}$ or equivalently, all elements of ${\displaystyle B}$ that also belong to ${\displaystyle A.}$^[2]

Notation and terminology

Intersection is written using the symbol "${\displaystyle \cap }$" between the terms; that is, in infix notation. For example:

The intersection of more than two sets (generalized intersection) can be written as: 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 ${\displaystyle A}$ and ${\displaystyle B,}$ denoted by ${\displaystyle A\cap B}$,^[3] is the set of all objects that are members of both the sets ${\displaystyle A}$ and ${\displaystyle B.}$ In symbols:

That is, ${\displaystyle x}$ is an element of the intersection ${\displaystyle A\cap B}$ if and only if ${\displaystyle x}$ is both an element of ${\displaystyle A}$ and an element of ${\displaystyle B.}$^[3]

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, ...}, because 9 is not prime.

Intersecting and disjoint sets

We say that ${\displaystyle A}$ intersects (meets) ${\displaystyle B}$ if there exists some ${\displaystyle x}$ that is an element of both ${\displaystyle A}$ and ${\displaystyle B,}$ in which case we also say that ${\displaystyle A}$ intersects (meets) ${\displaystyle B}$ at ${\displaystyle x}$. Equivalently, ${\displaystyle A}$ intersects ${\displaystyle B}$ if their intersection ${\displaystyle A\cap B}$ is an inhabited set, meaning that there exists some ${\displaystyle x}$ such that ${\displaystyle x\in A\cap B.}$

We say that ${\displaystyle A}$ and ${\displaystyle B}$ are disjoint if ${\displaystyle A}$ does not intersect ${\displaystyle B.}$ In plain language, they have no elements in common. ${\displaystyle A}$ and ${\displaystyle B}$ are disjoint if their intersection is empty, denoted ${\displaystyle A\cap B=\varnothing .}$

For example, the sets ${\displaystyle \{1,2\}}$ and ${\displaystyle \{3,4\}}$ 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 ${\displaystyle A,B,}$ and ${\displaystyle C,}$ one has

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

Intersection distributes over union and union distributes over intersection. That is, for any sets ${\displaystyle A,B,}$ and ${\displaystyle C,}$ one has

Inside a universe ${\displaystyle U,}$ one may define the complement ${\displaystyle A^{c}}$ of ${\displaystyle A}$ to be the set of all elements of ${\displaystyle U}$ not in ${\displaystyle A.}$ Furthermore, the intersection of ${\displaystyle A}$ and ${\displaystyle B}$ may be written as the complement of the union of their complements, derived easily from De Morgan's laws:

Arbitrary intersections

The most general notion is the intersection of an arbitrary nonempty collection of sets. If ${\displaystyle M}$ is a nonempty set whose elements are themselves sets, then ${\displaystyle x}$ is an element of the intersection of ${\displaystyle M}$ if and only if for every element ${\displaystyle A}$ of ${\displaystyle M,}$ ${\displaystyle x}$ is an element of ${\displaystyle A.}$ In symbols:

The notation for this last concept can vary considerably. Set theorists will sometimes write "${\displaystyle \bigcap M}$", while others will instead write "${\displaystyle {\bigcap }_{A\in M}A}$". The latter notation can be generalized to "${\displaystyle {\bigcap }_{i\in I}A_{i}}$", which refers to the intersection of the collection ${\displaystyle \left\{A_{i}:i\in I\right\}.}$ Here ${\displaystyle I}$ is a nonempty set, and ${\displaystyle A_{i}}$ is a set for every ${\displaystyle i\in I.}$

In the case that the index set ${\displaystyle I}$ is the set of natural numbers, notation analogous to that of an infinite product may be seen:

When formatting is difficult, this can also be written "${\displaystyle A_{1}\cap A_{2}\cap A_{3}\cap \cdots }$". This last example, an intersection of countably many sets, is actually very common; for an example, see the article on σ-algebras.

Nullary intersection

In the previous section, we excluded the case where ${\displaystyle M}$ was the empty set (${\displaystyle \varnothing }$). The reason is as follows: The intersection of the collection ${\displaystyle M}$ is defined as the set (see set-builder notation)

If ${\displaystyle M}$ is empty, there are no sets ${\displaystyle A}$ in ${\displaystyle M,}$ so the question becomes "which ${\displaystyle x}$'s satisfy the stated condition?" The answer seems to be every possible ${\displaystyle x}$. When ${\displaystyle M}$ 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),^[4] but in standard (ZF) set theory, the universal set does not exist.

However, when restricted to the context of subsets of a given fixed set ${\displaystyle X}$, the notion of the intersection of an empty collection of subsets of ${\displaystyle X}$ is well-defined. In that case, if ${\displaystyle M}$ is empty, its intersection is ${\displaystyle \bigcap M=\bigcap \varnothing =\{x\in X:x\in A{\text{ for all }}A\in \varnothing \}}$. Since all ${\displaystyle x\in X}$ vacuously satisfy the required condition, the intersection of the empty collection of subsets of ${\displaystyle X}$ is all of ${\displaystyle X.}$ In formulas, ${\displaystyle \bigcap \varnothing =X.}$ This matches the intuition that as collections of subsets become smaller, their respective intersections become larger; in the extreme case, the empty collection has an intersection equal to the whole underlying set.

Also, in type theory ${\displaystyle x}$ is of a prescribed type ${\displaystyle \tau ,}$ so the intersection is understood to be of type ${\displaystyle \mathrm {set} \ \tau }$ (the type of sets whose elements are in ${\displaystyle \tau }$), and we can define ${\displaystyle \bigcap _{A\in \emptyset }A}$ to be the universal set of ${\displaystyle \mathrm {set} \ \tau }$ (the set whose elements are exactly all terms of type ${\displaystyle \tau }$).

See also

• Algebra of sets – Identities and relationships involving sets
• Cardinality – Definition of the number of elements in a set
• Complement – Set of the elements not in a given subset
• Intersection (Euclidean geometry) – Shape formed from points common to other shapesPages displaying short descriptions of redirect targets
• Intersection graph – Graph representing intersections between given sets
• Intersection theory – Branch of algebraic geometry
• List of set identities and relations – Equalities for combinations of sets
• Logical conjunction – Logical connective AND
• MinHash – Data mining technique
• Naive set theory – Informal set theories
• Symmetric difference – Elements in exactly one of two sets
• Union – Set of elements in any of some sets

References

Further reading

• Devlin, K. J. (1993). The Joy of Sets: Fundamentals of Contemporary Set Theory (Second ed.). New York, NY: Springer-Verlag. ISBN 3-540-94094-4.
• Munkres, James R. (2000). "Set Theory and Logic". Topology (Second ed.). Upper Saddle River: Prentice Hall. ISBN 0-13-181629-2.
• Rosen, Kenneth (2007). "Basic Structures: Sets, Functions, Sequences, and Sums". Discrete Mathematics and Its Applications (Sixth ed.). Boston: McGraw-Hill. ISBN 978-0-07-322972-0.

External links

• Weisstein, Eric W. "Intersection". MathWorld.