Type  Set operation 

Field  Set theory 
Statement  The intersection of and is the set of elements that lie in both set and set . 
Symbolic statement 
In set theory, the intersection of two sets and denoted by ^{[1]} is the set containing all elements of that also belong to or equivalently, all elements of that also belong to ^{[2]}
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Intersection of Sets, Union of Sets and Venn Diagrams

Intersection and union of sets  Probability and Statistics  Khan Academy

Set Theory: Types of Sets, Unions and Intersections

Sets: Union and Intersection

Sets: Union, Intersection, Complement
Transcription
Notation and terminology
Intersection is written using the symbol "" 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 capitalsigma notation.
For an explanation of the symbols used in this article, refer to the table of mathematical symbols.
Definition
The intersection of two sets and denoted by ,^{[3]} is the set of all objects that are members of both the sets and In symbols:
That is, is an element of the intersection if and only if is both an element of and an element of ^{[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 intersects (meets) if there exists some that is an element of both and in which case we also say that intersects (meets) at . Equivalently, intersects if their intersection is an inhabited set, meaning that there exists some such that
We say that and are disjoint if does not intersect In plain language, they have no elements in common. and are disjoint if their intersection is empty, denoted
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 and one has
Thus the parentheses may be omitted without ambiguity: either of the above can be written as . Intersection is also commutative. That is, for any and one has The intersection of any set with the empty set results in the empty set; that is, that for any set , Also, the intersection operation is idempotent; that is, any set satisfies that . All these properties follow from analogous facts about logical conjunction.
Intersection distributes over union and union distributes over intersection. That is, for any sets and one has Inside a universe one may define the complement of to be the set of all elements of not in Furthermore, the intersection of and 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 is a nonempty set whose elements are themselves sets, then is an element of the intersection of if and only if for every element of is an element of In symbols:
The notation for this last concept can vary considerably. Set theorists will sometimes write "", while others will instead write "". The latter notation can be generalized to "", which refers to the intersection of the collection Here is a nonempty set, and is a set for every
In the case that the index set 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 "". This last example, an intersection of countably many sets, is actually very common; for an example, see the article on σalgebras.
Nullary intersection
In the previous section, we excluded the case where was the empty set (). The reason is as follows: The intersection of the collection is defined as the set (see setbuilder notation) If is empty, there are no sets in so the question becomes "which 's satisfy the stated condition?" The answer seems to be every possible . When 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 , the notion of the intersection of an empty collection of subsets of is welldefined. In that case, if is empty, its intersection is . Since all vacuously satisfy the required condition, the intersection of the empty collection of subsets of is all of In formulas, 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 is of a prescribed type so the intersection is understood to be of type (the type of sets whose elements are in ), and we can define to be the universal set of (the set whose elements are exactly all terms of type ).
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 shapes
 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
 ^ "Intersection of Sets". web.mnstate.edu. Archived from the original on 20200804. Retrieved 20200904.
 ^ "Stats: Probability Rules". People.richland.edu. Retrieved 20120508.
 ^ ^{a} ^{b} "Set Operations  Union  Intersection  Complement  Difference  Mutually Exclusive  Partitions  De Morgan's Law  Distributive Law  Cartesian Product". www.probabilitycourse.com. Retrieved 20200904.
 ^ Megginson, Robert E. (1998). "Chapter 1". An introduction to Banach space theory. Graduate Texts in Mathematics. Vol. 183. New York: SpringerVerlag. pp. xx+596. ISBN 0387984313.
Further reading
 Devlin, K. J. (1993). The Joy of Sets: Fundamentals of Contemporary Set Theory (Second ed.). New York, NY: SpringerVerlag. ISBN 3540940944.
 Munkres, James R. (2000). "Set Theory and Logic". Topology (Second ed.). Upper Saddle River: Prentice Hall. ISBN 0131816292.
 Rosen, Kenneth (2007). "Basic Structures: Sets, Functions, Sequences, and Sums". Discrete Mathematics and Its Applications (Sixth ed.). Boston: McGrawHill. ISBN 9780073229720.