Talìa lu còdici di Suttanzemi

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[[Image:Venn A subset B.svg|150px|thumb|right|Nu [[diaggramma di Euleru]] ca fa viriri<br> ''A'' comu suttanzemi di ''B'' e viciversa ''B'' comu supranzemi di ''A'']]
 
'Nta [[matimatica]], spiciarmenti 'nta [[tiurìa di li nzemi]], nu [[Nzemi (matimatica)|nzemi]] ''A'' enu nu '''suttanzemi''' di lu nzemi ''B'' si ''A'' eni "cuntinutu" rintra di ''B''.<br>
Ppi piaciri nutati ca ''A'' e ''B'' putissiru cuincìdiri.<br>
A rilazzioni di nu nzemi cu n'autru e ciamatu '''inclusioni'''.
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==Difinizziona==
Si ''A'' e ''B'' sunu nzemi di ogni [[ilimentu (matimatica)|ilimentu]] di ''A'' allura aviemu macari is also an element of ''B'', then:
:* ''A'' is a subset of (or is included in) ''B'', denoted by <math>A \subseteq B</math>,
:or equivalently
:* ''B'' is a '''superset''' of (or includes) ''A'', denoted by <math>B \supseteq A.</math>

If ''A'' is a subset of ''B'', but ''A'' is not [[equal (math)|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 <math>A\subsetneq B.</math>
:or equivalently
:* ''B'' is a proper superset of ''A''; this is written as <math>B\supsetneq A.</math>

For any set ''S'', the inclusion relation ⊆ is a [[partial order]] on the set 2<sup>''S''</sup> of all subsets of ''S'' (the [[power set]] of ''S'').
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==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''&nbsp;⊂&nbsp;''A''.

Other authors prefer to use the symbols ⊂ and ⊃ to indicate ''proper'' subset and superset, respectively, in place of <math>\subsetneq</math> and <math>\supsetneq.</math> This usage makes ⊆ and ⊂ analogous to ≤ and <. For example, if ''x''&nbsp;≤&nbsp;''y'' then ''x'' may be equal to ''y'', or maybe not, but if ''x''&nbsp;<&nbsp;''y'', then ''x'' definitely does not equal ''y'', but is strictly less than ''y''. Similarly, using the "⊂ means proper subset" convention, if ''A''&nbsp;⊆&nbsp;''B'', then ''A'' may or may not be equal to ''B'', but if ''A''&nbsp;⊂&nbsp;''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.
* The [[empty set]], denoted by &empty;, is also a subset of any given set ''X''. (This statement is [[vacuous truth|vacuously true]].) The empty set is always a proper subset, except of itself.
* The set {''x'': ''x'' is a [[prime number]] greater than 2000} is a proper subset of {''x'': ''x'' is an odd number greater than 1000}
* The set of [[natural number]]s is a proper subset of the set of [[rational number]]s and the set of points in a [[line segment]] is a proper subset of the set of points in a [[line (mathematics)|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#Infinite sets|Cardinality of infinite sets]]).

== Other properties of inclusion ==

Inclusion is the canonical [[partial order]] in the sense that every partially ordered set (''X'', <math>\preceq</math>) is [[isomorphic]] to some collection of sets ordered by inclusion. The [[ordinal 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<sup>''S''</sup> of a set ''S'', the inclusion partial order is (up to an [[order isomorphism]]) the [[Cartesian product]] of ''k'' = |''S''| (the [[cardinality]] of ''S'') copies of the partial order on {0,1} for which 0 &lt; 1. This can be illustrated by enumerating ''S'' = {''s''<sub>1</sub>, ''s''<sub>2</sub>, …, ''s''<sub>''k''</sub>} and associating with each subset ''T'' ⊆ ''S'' (which is to say with each element of 2<sup>''S''</sup>) the ''k''-tuple from {0,1}<sup>''k''</sup> of which the ''i''th coordinate is 1 if and only if ''s''<sub>''i''</sub> is a member of ''T''.

==References==
* {{cite book|author=Jech, Thomas|title=Set Theory|publisher=Springer-Verlag|year=2002|isbn=3-540-44085-2}}

 
[[Catigurìa:Basic concepts in set theory]]
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[[Catigurìa:Matimàtica]]
[[Catigurìa:Tiurìi]]

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