- Subset
In

mathematics , especially inset theory , a set "A" is a**subset**of a set "B" if "A" is "contained" inside "B". Notice that "A" and "B" may coincide. The relationship of one set being a subset of another is called**inclusion**.**Definitions**If "A" and "B" are sets and every element of "A" is also an element of "B", then::* "A" is a subset of (or is included in) "B", denoted by $A\; subseteq\; B$,:or equivalently:* "B" is a

**superset**of (or includes) "A", denoted by $B\; supseteq\; A.$If "A" is a subset of "B", but "A" is not equal to "B" (i.e. there exists at least one element of B not contained in "A"), then :* "A" is also a

**proper**(or**strict**) subset of "B"; this is written as $Asubsetneq\; B.$:or equivalently:* "B" is a proper superset of "A"; this is written as $Bsupsetneq\; A.$For any set "S", the inclusion relation ⊆ is a

partial order on the set 2^{"S"}of all subsets of "S" (thepower set of "S").**The symbols ⊂ and ⊃**Some authors use the symbols ⊂ and ⊃ to indicate "subset" and "superset" respectively, instead of the symbols ⊆ and ⊇, but with the same meaning. So for example, for these authors, it is true of every set "A" that "A" ⊂ "A".

Other authors prefer to use the symbols ⊂ and ⊃ to indicate "proper" subset and superset, respectively, in place of $subsetneq$ and $supsetneq.$ This usage makes ⊆ and ⊂ analogous to ≤ and <. For example, if "x" ≤ "y" then "x" may be equal to "y", or maybe not, but if "x" < "y", then "x" definitely does not equal "y", but is strictly less than "y". Similarly, using the "⊂ means proper subset" convention, if "A" ⊆ "B", then "A" may or may not be equal to "B", but if "A" ⊂ "B", then "A" is definitely not equal to "B".

**Examples*** The set {1, 2} is a proper subset of {1, 2, 3}.

* Any set is a subset of itself, but not a proper subset.

* Theempty set , written Unicode|∅, is also a subset of any given set "X". (This statement is vacuously true.) The empty set is always a proper subset, except of itself.

* The set {"x": "x" is aprime number greater than 2000} is a proper subset of {"x": "x" is an odd number greater than 1000}

* The set ofnatural number s is a proper subset of the set ofrational number s and the set of points in aline segment is a proper subset of the set of points in a line. These are counter-intuitive examples in which both the part and the whole are infinite, and the part has the same number of elements as the whole (see Cardinality of infinite sets).**Other properties of inclusion**Inclusion is the canonical

partial order in the sense that every partially ordered set ("X", $preceq$) isisomorphic to some collection of sets ordered by inclusion. Theordinal number s are a simple example—if each ordinal "n" is identified with the set ["n"] of all ordinals less than or equal to "n", then "a" ≤ "b" if and only if ["a"] ⊆ ["b"] .For the

power set 2^{"S"}of a set "S", the inclusion partial order is (up to anorder isomorphism ) theCartesian product of "k" = |"S"| (thecardinality of "S") copies of the partial order on {0,1} for which 0 < 1. This can be illustrated by enumerating "S" = {"s"_{1}, "s"_{2}, …, "s"_{"k"}} and associating with each subset "T" ⊆ "S" (which is to say with each element of 2^{"S"}) the "k"-tuple from {0,1}^{"k"}of which the "i"th coordinate is 1 if and only if "s"_{"i"}is a member of "T".**References***

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Subset**— Pays d’origine États Unis Genre musical Rock Années d activité 1999 … Wikipédia en Français**subset**— [sub′set΄] n. a mathematical set in which every element in the set is also contained in a larger set or in an equal set [the even numbers are a subset of the whole numbers] … English World dictionary**subset**— (n.) subordinate set, 1902, from SUB (Cf. sub ) + SET (Cf. set) (n.) … Etymology dictionary**subset**— ► NOUN 1) a part of a larger group of related things. 2) Mathematics a set of which all the elements are contained in another set … English terms dictionary**subset**— noun /ˈsʌbˌsɛt/ a) With respect to another set, a set such that each of its elements is also an element of the other set. The set of integers is a subset of the set of reals. b) A group of things or people, all of which are in a specified larger… … Wiktionary**subset**— UK [ˈsʌbˌset] / US noun [countable] Word forms subset : singular subset plural subsets a small group of people or things that is a part of a larger group Each patient is likely to show only a subset of the symptoms listed … English dictionary**subset**— sub|set [ˈsʌbset] n a group of people or things that is part of a larger group of people or things subset of ▪ a small subset of the city s immigrant population … Dictionary of contemporary English**subset**— noun Date: 1902 1. a set each of whose elements is an element of an inclusive set 2. division, portion < a subset of our community > … New Collegiate Dictionary**subset**— [[t]sʌ̱bset[/t]] subsets N COUNT: oft N of n A subset of a group of things is a smaller number of things that belong together within that group. ...subsets of the population such as men, women, ethnic groups, etc … English dictionary**subset**— /ˈsʌbsɛt/ (say subset) noun a set whose elements belong, with others, to a given, larger set; a subordinate set … Australian English dictionary