Intersection (set theory)

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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]

Notation and terminology[edit]

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[edit]

Intersection of three sets:
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

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:

  • 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 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[edit]

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[edit]

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[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.

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.

See also[edit]

References[edit]

  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[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]