Intersection (set theory)

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]

The intersection of two sets and represented by circles. is in red.
TypeSet operation
FieldSet theory
StatementThe intersection of and is the set of elements that lie in both set and set .
Symbolic statement

Notation and terminology edit

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 capital-sigma notation.

For an explanation of the symbols used in this article, refer to the table of mathematical symbols.

Definition edit

Intersection of three sets:
Intersections of the unaccented modern Greek, Latin, and Cyrillic scripts, considering only the shapes of the letters and ignoring their pronunciation
Example of an intersection with sets

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 edit

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 edit

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 edit

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 edit

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.

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),[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 well-defined. 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 edit

References edit

  1. ^ "Intersection of Sets". Archived from the original on 2020-08-04. Retrieved 2020-09-04.
  2. ^ "Stats: Probability Rules". Retrieved 2012-05-08.
  3. ^ a b "Set Operations | Union | Intersection | Complement | Difference | Mutually Exclusive | Partitions | De Morgan's Law | Distributive Law | Cartesian Product". Retrieved 2020-09-04.
  4. ^ Megginson, 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.

Further reading edit

  • 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 edit