Difference between revisions of "Intersection"
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==Origin== | ==Origin== | ||
− | [[Latin]] ''intersectiōn-em'' ([ | + | [[Latin]] ''intersectiōn-em'' ([https://en.wikipedia.org/wiki/Vitruvius Vitruvius]), n. of [[action]] from ''intersecāre'' |
− | *[ | + | *[https://en.wikipedia.org/wiki/16th_century 1559] |
==Definitions== | ==Definitions== | ||
*1: The [[action]] or [[fact]] of intersecting or crossing | *1: The [[action]] or [[fact]] of intersecting or crossing | ||
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:b. [[Geometry]], the [[point]] (or [[line]]) of intersection; the point common to two lines or a line and a [[surface]] (or the line common to two surfaces) which intersect. | :b. [[Geometry]], the [[point]] (or [[line]]) of intersection; the point common to two lines or a line and a [[surface]] (or the line common to two surfaces) which intersect. | ||
*3:a. [[Logic]]. The [[relation]] of two classes that intersect, i.e. each of which partly includes and partly excludes the other. | *3:a. [[Logic]]. The [[relation]] of two classes that intersect, i.e. each of which partly includes and partly excludes the other. | ||
− | :b. Logic and [[Mathematics]]. The [ | + | :b. Logic and [[Mathematics]]. The [https://en.wikipedia.org/wiki/Set set] which comprises all the elements common to [[two]] or more given sets, and no others; also, the operation of forming such a set. |
==Description== | ==Description== | ||
In [[mathematics]], the intersection (denoted as ∩) of two sets A and B is the set that contains all [[elements]] of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. | In [[mathematics]], the intersection (denoted as ∩) of two sets A and B is the set that contains all [[elements]] of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. | ||
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The intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}. | 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 [ | + | The [[number]] [[9]] is not in the intersection of the set of [https://en.wikipedia.org/wiki/Prime_number prime numbers] {2, 3, 5, 7, 11, …} and the set of [https://en.wikipedia.org/wiki/Odd_numbers odd numbers] {1, 3, 5, 7, 9, 11, …}. |
If the intersection of two sets A and B is empty, that is they have no elements in common, then they are said to be disjoint, denoted: A ∩ B = ∅. For example the sets {1, 2} and {3, 4} are disjoint, written {1, 2} ∩ {3, 4} = ∅. | If the intersection of two sets A and B is empty, that is they have no elements in common, then they are said to be disjoint, denoted: A ∩ B = ∅. For example the sets {1, 2} and {3, 4} are disjoint, written {1, 2} ∩ {3, 4} = ∅. | ||
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More generally, one can take the intersection of several sets at once. The intersection of A, B, C, and D, for example, is A ∩ B ∩ C ∩ D = A ∩ (B ∩ (C ∩ D)). Intersection is an associative operation; thus, A ∩ (B ∩ C) = (A ∩ B) ∩ C. | More generally, one can take the intersection of several sets at once. The intersection of A, B, C, and D, for example, is A ∩ B ∩ C ∩ D = A ∩ (B ∩ (C ∩ D)). Intersection is an associative operation; thus, A ∩ (B ∩ C) = (A ∩ B) ∩ C. | ||
− | If the sets A and B are closed under complement then the intersection of A and B may be written as the complement of the union of their complements, derived easily from [ | + | If the sets A and B are closed under complement then the intersection of A and B may be written as the complement of the union of their complements, derived easily from [https://en.wikipedia.org/wiki/De_Morgan%27s_laws De Morgan's laws]: A ∩ B = (Ac ∪ Bc)c[https://en.wikipedia.org/wiki/Intersection_%28set_theory%29] |
[[Category: Mathematics]] | [[Category: Mathematics]] |
Latest revision as of 01:24, 13 December 2020
Origin
Latin intersectiōn-em (Vitruvius), n. of action from intersecāre
Definitions
- 1: The action or fact of intersecting or crossing
- 2:a. The place where two things intersect or cross, spec.cross-road
- b. Geometry, the point (or line) of intersection; the point common to two lines or a line and a surface (or the line common to two surfaces) which intersect.
- 3:a. Logic. The relation of two classes that intersect, i.e. each of which partly includes and partly excludes the other.
- b. Logic and Mathematics. The set which comprises all the elements common to two or more given sets, and no others; also, the operation of forming such a set.
Description
In mathematics, the intersection (denoted as ∩) of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.
The intersection of A and B is written "A ∩ B". Formally:
- x ∈ A ∩ B if and only if
- x ∈ A and
- x ∈ B.
- 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, …}.
If the intersection of two sets A and B is empty, that is they have no elements in common, then they are said to be disjoint, denoted: A ∩ B = ∅. For example the sets {1, 2} and {3, 4} are disjoint, written {1, 2} ∩ {3, 4} = ∅.
More generally, one can take the intersection of several sets at once. The intersection of A, B, C, and D, for example, is A ∩ B ∩ C ∩ D = A ∩ (B ∩ (C ∩ D)). Intersection is an associative operation; thus, A ∩ (B ∩ C) = (A ∩ B) ∩ C.
If the sets A and B are closed under complement then 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: A ∩ B = (Ac ∪ Bc)c[1]