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Intersection (set theory)

From Wikipedia, the free encyclopedia

The intersection of two sets  A {\displaystyle A}  and  B , {\displaystyle B,}  represented by circles.  A ∩ B {\displaystyle A\cap B}  is in red.
The intersection of two sets and represented by circles. is in red.

In mathematics, the intersection of two sets and denoted by [1][2] is the set containing all elements of that also belong to or equivalently, all elements of that also belong to [3]

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:[1]
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

Intersection of three sets:   A ∩ B ∩ C {\displaystyle ~A\cap B\cap C}
Intersection of three sets:
Intersections of the Greek, Latin and Russian alphabet, considering only the shapes of the letters and ignoring their pronunciation
Intersections of the Greek, Latin and Russian alphabet, considering only the shapes of the letters and ignoring their pronunciation
Example of an intersection with sets
Example of an intersection with sets

The intersection of two sets and denoted by [1][4] 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 [4]

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.

Intersection is an associative operation; that is, for any sets and one has Intersection is also commutative; for any and one has It thus makes sense to talk about intersections of multiple sets. The intersection of for example, is unambiguously written

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:

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.

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 σ-algebras.

Nullary intersection

Conjunctions of the arguments in parenthesesThe conjunction of no argument is the tautology (compare: empty product); accordingly the intersection of no set is the universe.
Conjunctions of the arguments in parentheses

The conjunction of no argument is the tautology (compare: empty product); accordingly the intersection of no set is the universe.

Note that 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 set-builder 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),[5] but in standard (ZFC) set theory, the universal set does not exist.

In type theory however, 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

References

  1. ^ a b c "Comprehensive List of Set Theory Symbols". Math Vault. 2020-04-11. Retrieved 2020-09-04.
  2. ^ "Intersection of Sets". web.mnstate.edu. Retrieved 2020-09-04.
  3. ^ "Stats: Probability Rules". People.richland.edu. Retrieved 2012-05-08.
  4. ^ a b "Set Operations | Union | Intersection | Complement | Difference | Mutually Exclusive | Partitions | De Morgan's Law | Distributive Law | Cartesian Product". www.probabilitycourse.com. Retrieved 2020-09-04.
  5. ^ Megginson, Robert E. (1998), "Chapter 1", An introduction to Banach space theory, Graduate Texts in Mathematics, 183, New York: Springer-Verlag, pp. xx+596, ISBN 0-387-98431-3

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

This page was last edited on 5 September 2021, at 04:19
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