Intersection (set theory)

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The intersection of two sets
A
{\displaystyle A}
and
B
{\displaystyle B}
, represented by circles.
A
[?]
B
{\displaystyle A\cap B}
is in red. Venn0001.svg
The intersection of two sets and , represented by circles. is in red.

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

Contents

Notation and terminology

Intersection is written using the sign "∩" 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} Venn 0000 0001.svg
Intersection of three sets:
Intersections of the Greek, Latin and Russian alphabet, considering only the shapes of the letters and ignoring their pronunciation Venn diagram gr la ru.svg
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 PolygonsSetIntersection.svg
Example of an intersection with sets

The intersection of two sets A and B, denoted by AB, [1] [4] is the set of all objects that are members of both the sets A and B. In symbols,

That is, x is an element of the intersection AB, if and only if x is both an element of A and an element of B. [4]

For example:

Intersection is an associative operation; that is, for any sets A, B, and C, one has A ∩ (BC) = (AB) ∩ C. Intersection is also commutative; for any A and B, one has AB = BA. It thus makes sense to talk about intersections of multiple sets. The intersection of A, B, C, and D, for example, is unambiguously written ABCD.

Inside a universe U, one may define the complement Ac 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:
AB = (AcBc)c

Intersecting and disjoint sets

We say that A intersects (meets) B at an element x if x belongs to A and B. We say that A intersects (meets) B if A intersects B at some element. A intersects B if their intersection is inhabited.

We say that A and B are disjoint 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 .

For example, the sets {1, 2} and {3, 4} 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 M is a nonempty set whose elements are themselves sets, then x is an element of the intersection of M if and only if for every element A of M, x is an element of A. In symbols:

The notation for this last concept can vary considerably. Set theorists will sometimes write "⋂M", while others will instead write "⋂AM A". The latter notation can be generalized to "⋂iI Ai", which refers to the intersection of the collection {Ai : i  I}. Here I is a nonempty set, and Ai is a set for every i in I.

In the case that the index set 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 "A1 A2 A3 ∩ ...". 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 parentheses

The conjunction of no argument is the tautology (compare: empty product); accordingly the intersection of no set is the universe. Multigrade operator AND.svg
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 M was the empty set (∅). The reason is as follows: The intersection of the collection M is defined as the set (see set-builder notation)

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 every possible x. When 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), [5] but in standard (ZFC) set theory, the universal set does not exist.

In type theory however, x 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

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References

  1. 1 2 3 "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. 1 2 "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