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

• Notation and terminology

• Definition

• Intersecting and disjoint sets

• Algebraic properties

• Arbitrary intersections

• Nullary intersection

• See also

• References

• Further reading

• External links

Intersection is written using the symbol " ∩ ∩ {\displaystyle \cap } " between the terms; that is, in infix notation. For example: { 1 , 2 , 3 } ∩ ∩ { 2 , 3 , 4 } = { 2 , 3 } {\displaystyle \{1,2,3\}\cap \{2,3,4\}=\{2,3\}} { 1 , 2 , 3 } ∩ ∩ { 4 , 5 , 6 } = ∅ ∅ {\displaystyle \{1,2,3\}\cap \{4,5,6\}=\varnothing } Z ∩ ∩ N = N {\displaystyle \mathbb {Z} \cap \mathbb {N} =\mathbb {N} } { x ∈ ∈ R : x 2 = 1 } ∩ ∩ N = { 1 } {\displaystyle \{x\in \mathbb {R} :x^{2}=1\}\cap \mathbb {N} =\{1\}} The intersection of more than two sets (generalized intersection) can be written as: ⋂ ⋂ i = 1 n A i {\displaystyle \bigcap _{i=1}^{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.

The intersection of two sets A {\displaystyle A} and B , {\displaystyle B,} denoted by A ∩ ∩ B {\displaystyle A\cap B} ,cite-ref-1-3-0[3] is the set of all objects that are members of both the sets A {\displaystyle A} and B . {\displaystyle B.} In symbols: A ∩ ∩ B = { x : x ∈ ∈ A and x ∈ ∈ B } . {\displaystyle A\cap B=\{x:x\in A{\text{ and }}x\in B\}.}

That is, x {\displaystyle x} is an element of the intersection A ∩ ∩ B {\displaystyle A\cap B} if and only if x {\displaystyle x} is both an element of A {\displaystyle A} and an element of B . {\displaystyle B.} cite-ref-1-3-1[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.

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

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

For example, the sets { 1 , 2 } {\displaystyle \{1,2\}} and { 3 , 4 } {\displaystyle \{3,4\}} are disjoint, while the set of even numbers intersects the set of multiples of 3 at the multiples of 6.

Binary intersection is an associative operation; that is, for any sets A , B , {\displaystyle A,B,} and C , {\displaystyle C,} one has

A ∩ ∩ ( B ∩ ∩ C ) = ( A ∩ ∩ B ) ∩ ∩ C . {\displaystyle 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 ∩ ∩ B ∩ ∩ C {\displaystyle A\cap B\cap C} . Intersection is also commutative. That is, for any A {\displaystyle A} and B , {\displaystyle B,} one has A ∩ ∩ B = B ∩ ∩ A . {\displaystyle 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 {\displaystyle A} , A ∩ ∩ ∅ ∅ = ∅ ∅ {\displaystyle A\cap \varnothing =\varnothing } Also, the intersection operation is idempotent; that is, any set A {\displaystyle A} satisfies that A ∩ ∩ A = A {\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 A , B , {\displaystyle A,B,} and C , {\displaystyle C,} one has A ∩ ∩ ( B ∪ ∪ C ) = ( A ∩ ∩ B ) ∪ ∪ ( A ∩ ∩ C ) A ∪ ∪ ( B ∩ ∩ C ) = ( A ∪ ∪ B ) ∩ ∩ ( A ∪ ∪ C ) {\displaystyle {\begin{aligned}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{aligned}}} Inside a universe U , {\displaystyle U,} one may define the complement A c {\displaystyle A^{c}} of A {\displaystyle A} to be the set of all elements of U {\displaystyle U} not in A . {\displaystyle A.} Furthermore, the intersection of A {\displaystyle A} and B {\displaystyle 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 {\displaystyle A\cap B=\left(A^{c}\cup B^{c}\right)^{c}}

The most general notion is the intersection of an arbitrary nonempty collection of sets. If M {\displaystyle M} is a nonempty set whose elements are themselves sets, then x {\displaystyle x} is an element of the intersection of M {\displaystyle M} if and only if for every element A {\displaystyle A} of M , {\displaystyle M,} x {\displaystyle x} is an element of A . {\displaystyle A.} In symbols: ( x ∈ ∈ ⋂ ⋂ A ∈ ∈ M A ) ⇔ ⇔ ( ∀ ∀ A ∈ ∈ M , x ∈ ∈ A ) . {\displaystyle \left(x\in \bigcap _{A\in M}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 " ⋂ ⋂ M {\displaystyle \bigcap M} ", while others will instead write " ⋂ ⋂ A ∈ ∈ M A {\displaystyle {\bigcap }_{A\in M}A} ". The latter notation can be generalized to " ⋂ ⋂ i ∈ ∈ I A i {\displaystyle {\bigcap }_{i\in I}A_{i}} ", which refers to the intersection of the collection { A i : i ∈ ∈ I } . {\displaystyle \left\{A_{i}:i\in I\right\}.} Here I {\displaystyle I} is a nonempty set, and A i {\displaystyle A_{i}} is a set for every i ∈ ∈ I . {\displaystyle i\in I.}

In the case that the index set I {\displaystyle I} is the set of natural numbers, notation analogous to that of an infinite product may be seen: ⋂ ⋂ i = 1 ∞ ∞ A i . {\displaystyle \bigcap _{i=1}^{\infty }A_{i}.}

When formatting is difficult, this can also be written " A 1 ∩ ∩ A 2 ∩ ∩ A 3 ∩ ∩ ⋯ ⋯ {\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.

In the previous section, we excluded the case where M {\displaystyle M} was the empty set ( ∅ ∅ {\displaystyle \varnothing } ). The reason is as follows: The intersection of the collection M {\displaystyle M} is defined as the set (see set-builder notation) ⋂ ⋂ A ∈ ∈ M A = { x : for all A ∈ ∈ M , x ∈ ∈ A } . {\displaystyle \bigcap _{A\in M}A=\{x:{\text{ for all }}A\in M,x\in A\}.} If M {\displaystyle M} is empty, there are no sets A {\displaystyle A} in M , {\displaystyle M,} so the question becomes "which x {\displaystyle x} 's satisfy the stated condition?" The answer seems to be every possible x {\displaystyle x} . When M {\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),cite-ref-4[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 X {\displaystyle X} , the notion of the intersection of an empty collection of subsets of X {\displaystyle X} is well-defined. In that case, if M {\displaystyle M} is empty, its intersection is ⋂ ⋂ M = ⋂ ⋂ ∅ ∅ = { x ∈ ∈ X : x ∈ ∈ A for all A ∈ ∈ ∅ ∅ } {\displaystyle \bigcap M=\bigcap \varnothing =\{x\in X:x\in A{\text{ for all }}A\in \varnothing \}} . Since all x ∈ ∈ X {\displaystyle x\in X} vacuously satisfy the required condition, the intersection of the empty collection of subsets of X {\displaystyle X} is all of X . {\displaystyle X.} In formulas, ⋂ ⋂ ∅ ∅ = X . {\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 x {\displaystyle x} is of a prescribed type τ τ , {\displaystyle \tau ,} so the intersection is understood to be of type s e t τ τ {\displaystyle \mathrm {set} \ \tau } (the type of sets whose elements are in τ τ {\displaystyle \tau } ), and we can define ⋂ ⋂ A ∈ ∈ ∅ ∅ A {\displaystyle \bigcap _{A\in \emptyset }A} to be the universal set of s e t τ τ {\displaystyle \mathrm {set} \ \tau } (the set whose elements are exactly all terms of type τ τ {\displaystyle \tau } ).

• Algebra of sets – Identities and relationships involving sets

• Cardinality – Size of a set in mathematics

• 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

cite-note-11. ↑ "Intersection of Sets". web.mnstate.edu. Archived from the original on 2020-08-04. Retrieved 2020-09-04.

cite-note-22. ↑ "Stats: Probability Rules". People.richland.edu. Retrieved 2012-05-08.

cite-note-1-33. ↑ "Set Operations | Union | Intersection | Complement | Difference | Mutually Exclusive | Partitions | De Morgan's Law | Distributive Law | Cartesian Product". www.probabilitycourse.com. Retrieved 2020-09-04.

cite-note-44. ↑ citerefmegginson1998Megginson, Robert E. (1998). "Chapter 1". An introduction to Banach space theory. Graduate Texts in Mathematics. Vol. 183. New York: Springer-Verlag. pp. xx+596. ISBN 0-387-98431-3.

• citerefdevlin1993Devlin, K. J. (1993). The Joy of Sets: Fundamentals of Contemporary Set Theory (Second ed.). New York, NY: Springer-Verlag. ISBN 3-540-94094-4.

• citerefmunkres2000Munkres, James R. (2000). "Set Theory and Logic". Topology (Second ed.). Upper Saddle River: Prentice Hall. ISBN 0-13-181629-2.

• citerefrosen2007Rosen, 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.

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• reference-mathworld-intersectionciterefweissteinWeisstein, Eric W. "Intersection". MathWorld.