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https:///mymemo.xyz/wiki/api.php?action=feedcontributions&user=182.171.246.130&feedformat=atom 2024-05-17T14:12:54Z 利用者の投稿記録 MediaWiki 1.31.0 2016-01-18T08:17:38Z

<p>182.171.246.130: </p> <hr /> <div>[[集合]] &#039;&#039;A&#039;&#039; が集合 &#039;&#039;B&#039;&#039; の&#039;&#039;&#039;部分集合&#039;&#039;&#039;（ぶぶんしゅうごう、&lt;em lang=&quot;en&quot;&gt;subset&lt;/em&gt;; 下位集合）であるとは、&#039;&#039;A&#039;&#039; が &#039;&#039;B&#039;&#039; の一部（あるいは全部）の要素だけからなることである。&#039;&#039;A&#039;&#039; が &#039;&#039;B&#039;&#039; の一部分であるという意味で部分集合という。二つの集合の一方が他方の部分集合であるとき、この二つの集合の間に&#039;&#039;&#039;包含関係&#039;&#039;&#039;があるという。<br /> [[Image:Venn A subset B.svg|thumb|right|真部分集合の[[オイラー図]]による視覚化]]<br /> <br /> == 定義 ==<br /> 集合 &#039;&#039;A&#039;&#039; の要素はすべて集合 &#039;&#039;B&#039;&#039; の要素でもあるとき、すなわち、<br /> :&lt;math&gt;\forall x (x\in A \rightarrow x\in B)&lt;/math&gt;<br /> が成り立つとき、&#039;&#039;A&#039;&#039; は &#039;&#039;B&#039;&#039; の&#039;&#039;&#039;部分集合&#039;&#039;&#039;であるといい、<br /> :&lt;math&gt;A\sube B&lt;/math&gt;<br /> で表す。&#039;&#039;A&#039;&#039; が &#039;&#039;B&#039;&#039; の部分集合であることを、「&#039;&#039;A&#039;&#039; は &#039;&#039;B&#039;&#039; に（部分集合として）含まれる（{{lang|en|contained}}; 包含される）」、「&#039;&#039;A&#039;&#039; は &#039;&#039;B&#039;&#039; に包まれる（{{lang|en|included}}; 包摂あるいは内包される）」などということもある。またこのとき、&#039;&#039;B&#039;&#039; は &#039;&#039;A&#039;&#039; の&#039;&#039;&#039;上位集合&#039;&#039;&#039;（じょういしゅうごう、{{lang|en|superset}}; スーパーセット）であるということもある。&#039;&#039;A&#039;&#039; が集合 &#039;&#039;B&#039;&#039; の要素であることも「&#039;&#039;A&#039;&#039; が &#039;&#039;B&#039;&#039; に含まれる」と表現している文献が存在するため、単に「&#039;&#039;A&#039;&#039; が &#039;&#039;B&#039;&#039; に含まれる」と言ったときは、&#039;&#039;A&#039;&#039; &amp;isin; &#039;&#039;B&#039;&#039; と &#039;&#039;A&#039;&#039; &amp;sube; &#039;&#039;B&#039;&#039; のどちらの意味で用いられているのかを文脈から判断することが必要である。<br /> <br /> 定義から任意の集合 &#039;&#039;B&#039;&#039; は自分自身の部分集合である。&#039;&#039;B&#039;&#039; 以外の集合で &#039;&#039;B&#039;&#039; の部分集合であるようなものは、&#039;&#039;B&#039;&#039; の&#039;&#039;&#039;真部分集合&#039;&#039;&#039;（しんぶぶんしゅうごう、{{lang|en|proper subset}}）あるいは&#039;&#039;&#039;狭義&#039;&#039;&#039;（{{lang|en|strict}}; 強い意味で）&#039;&#039;&#039;の部分集合&#039;&#039;&#039;と呼ばれる。すなわち、集合 &#039;&#039;A&#039;&#039; が集合 &#039;&#039;B&#039;&#039; の真部分集合であるとは、&#039;&#039;A&#039;&#039; &amp;sube; &#039;&#039;B&#039;&#039; かつ &#039;&#039;A&#039;&#039; &amp;ne; &#039;&#039;B&#039;&#039; が成り立つことである。&#039;&#039;A&#039;&#039; が &#039;&#039;B&#039;&#039; の真部分集合であることを<br /> :&lt;math&gt;A\sub B&lt;/math&gt;<br /> で表す。<br /> <br /> == 記法に関する注意 ==<br /> &lt;div style=&quot;float:right&quot;&gt;<br /> {| class=wikitable style=&quot;text-align:center; margin:1em;&quot;<br /> ! 部分集合 !! 真部分集合<br /> |-<br /> | &#039;&#039;A&#039;&#039; &amp;sube; &#039;&#039;B&#039;&#039; || &#039;&#039;A&#039;&#039; &amp;sub; &#039;&#039;B&#039;&#039;<br /> |-<br /> | &#039;&#039;A&#039;&#039; &amp;sub; &#039;&#039;B&#039;&#039; || &#039;&#039;A&#039;&#039; ⊊ &#039;&#039;B&#039;&#039;<br /> |- <br /> | &#039;&#039;A&#039;&#039; &amp;sube; &#039;&#039;B&#039;&#039; || &#039;&#039;A&#039;&#039; &amp;sube; &#039;&#039;B&#039;&#039; かつ &#039;&#039;A&#039;&#039; &amp;ne; &#039;&#039;B&#039;&#039;<br /> |- <br /> | &#039;&#039;A&#039;&#039; &amp;sub; &#039;&#039;B&#039;&#039; || &#039;&#039;A&#039;&#039; &amp;sub; &#039;&#039;B&#039;&#039; かつ &#039;&#039;A&#039;&#039; &amp;ne; &#039;&#039;B&#039;&#039;<br /> |- <br /> | &#039;&#039;A&#039;&#039; &amp;sube; &#039;&#039;B&#039;&#039; || &#039;&#039;A&#039;&#039; ⊊ &#039;&#039;B&#039;&#039;<br /> |}&lt;/div&gt;<br /> <br /> 上の定義においては、&#039;&#039;A&#039;&#039; が &#039;&#039;B&#039;&#039; の部分集合であることを &#039;&#039;A&#039;&#039; &amp;sube; &#039;&#039;B&#039;&#039; で表し、&#039;&#039;A&#039;&#039; が &#039;&#039;B&#039;&#039; の真部分集合であることを &#039;&#039;A&#039;&#039; &amp;sub; &#039;&#039;B&#039;&#039; で表した。実際、包含関係を[[順序集合|順序]]と見なせば、&#039;&#039;x&#039;&#039; &amp;le; &#039;&#039;y&#039;&#039; かつ &#039;&#039;x&#039;&#039; &amp;ne; &#039;&#039;y&#039;&#039; のとき &#039;&#039;x&#039;&#039; &amp;lt; &#039;&#039;y&#039;&#039; と記す慣例の類似で、&#039;&#039;A&#039;&#039; &amp;sub; &#039;&#039;B&#039;&#039; を &#039;&#039;A&#039;&#039; &amp;sube; &#039;&#039;B&#039;&#039; かつ &#039;&#039;A&#039;&#039; &amp;ne; &#039;&#039;B&#039;&#039; であるという意味で用いることは自然である。しかし、これとは異なった記法もいくつか存在し、統一されていない。例えば、&#039;&#039;A&#039;&#039; が &#039;&#039;B&#039;&#039; の部分集合であることを &#039;&#039;A&#039;&#039; &amp;sub; &#039;&#039;B&#039;&#039; で表し、&#039;&#039;A&#039;&#039; が &#039;&#039;B&#039;&#039; の真部分集合であることを &#039;&#039;A&#039;&#039; ⊊ &#039;&#039;B&#039;&#039; で表すという流儀もあり、この記法も広く用いられている。この他にも、部分集合には &amp;sube; を用い、真部分集合には &amp;sub; かつ &amp;ne; を用いるなど、さまざまなバリエーションがある（右表を参照）。部分集合であることを表すのに &amp;sube; 、真部分集合であることを表すのに ⊊ を用いると誤解を招く恐れがなくなるため、この記法も好まれる。<br /> <br /> == 基本的な性質 ==<br /> 以下、&#039;&#039;A&#039;&#039;, &#039;&#039;B&#039;&#039;, &#039;&#039;C&#039;&#039; を集合とする。<br /> * &#039;&#039;A&#039;&#039; = &#039;&#039;B&#039;&#039; と &#039;&#039;A&#039;&#039; &amp;sube; &#039;&#039;B&#039;&#039; かつ &#039;&#039;B&#039;&#039; &amp;sube; &#039;&#039;A&#039;&#039; は同値である（[[集合#外延性の原理|外延性の原理]]）。<br /> * [[空集合]] &amp;empty; はすべての集合の部分集合である。<br /> * &#039;&#039;A&#039;&#039; ⊆ &#039;&#039;A&#039;&#039; 。<br /> * &#039;&#039;A&#039;&#039; ⊆ &#039;&#039;B&#039;&#039; かつ &#039;&#039;B&#039;&#039; ⊆ &#039;&#039;C&#039;&#039; ならば &#039;&#039;A&#039;&#039; ⊆ &#039;&#039;C&#039;&#039; である。<br /> * &#039;&#039;A&#039;&#039; ⊆ &#039;&#039;A&#039;&#039; ∪ &#039;&#039;B&#039;&#039; 。<br /> * &#039;&#039;A&#039;&#039; ⊆ &#039;&#039;B&#039;&#039; ならば &#039;&#039;A&#039;&#039; ∪ &#039;&#039;C&#039;&#039; ⊆ &#039;&#039;B&#039;&#039; ∪ &#039;&#039;C&#039;&#039; 。<br /> * &#039;&#039;A&#039;&#039; ⊆ &#039;&#039;C&#039;&#039; かつ &#039;&#039;B&#039;&#039; ⊆ &#039;&#039;C&#039;&#039; ならば &#039;&#039;A&#039;&#039; ∪ &#039;&#039;B&#039;&#039; ⊆ &#039;&#039;C&#039;&#039; 。<br /> * &#039;&#039;A&#039;&#039; ∩ &#039;&#039;B&#039;&#039; ⊆ &#039;&#039;A&#039;&#039; 。<br /> * &#039;&#039;A&#039;&#039; ⊆ &#039;&#039;B&#039;&#039; ならば &#039;&#039;A&#039;&#039; ∩ &#039;&#039;C&#039;&#039; ⊆ &#039;&#039;B&#039;&#039; ∩ &#039;&#039;C&#039;&#039; 。<br /> * &#039;&#039;A&#039;&#039; ⊆ &#039;&#039;B&#039;&#039; かつ &#039;&#039;A&#039;&#039; ⊆ &#039;&#039;C&#039;&#039; ならば &#039;&#039;A&#039;&#039; ⊆ &#039;&#039;B&#039;&#039; ∩ &#039;&#039;C&#039;&#039; 。<br /> * &#039;&#039;A&#039;&#039; - &#039;&#039;B&#039;&#039; ⊆ &#039;&#039;A&#039;&#039; 。<br /> * &#039;&#039;A&#039;&#039; ⊆ &#039;&#039;B&#039;&#039; ならば &#039;&#039;A&#039;&#039; - &#039;&#039;C&#039;&#039; ⊆ &#039;&#039;B&#039;&#039; - &#039;&#039;C&#039;&#039; 。<br /> * &#039;&#039;A&#039;&#039; ⊆ &#039;&#039;B&#039;&#039; かつ &#039;&#039;A&#039;&#039; ⊆ &#039;&#039;C&#039;&#039; &lt;sup&gt;&#039;&#039;C&#039;&#039;&lt;/sup&gt; ならば &#039;&#039;A&#039;&#039; ⊆ &#039;&#039;B&#039;&#039; - &#039;&#039;C&#039;&#039; 。<br /> * 以下は[[同値]]である：<br /> **&#039;&#039;A&#039;&#039; ⊆ &#039;&#039;B&#039;&#039; 。<br /> **&#039;&#039;A&#039;&#039; ∩ &#039;&#039;B&#039;&#039; = &#039;&#039;A&#039;&#039; 。<br /> **&#039;&#039;A&#039;&#039; ∪ &#039;&#039;B&#039;&#039; = &#039;&#039;B&#039;&#039; 。<br /> **&#039;&#039;A&#039;&#039; − &#039;&#039;B&#039;&#039; = &amp;empty; 。<br /> * &#039;&#039;A&#039;&#039; と &#039;&#039;B&#039;&#039; がともに &#039;&#039;U&#039;&#039; の部分集合のとき、&#039;&#039;A&#039;&#039; &amp;sube; &#039;&#039;B&#039;&#039; と &#039;&#039;U&#039;&#039; - &#039;&#039;B&#039;&#039; ⊆ &#039;&#039;U&#039;&#039; - &#039;&#039;A&#039;&#039; は同値である。<br /> <br /> == 関連項目 ==<br /> *[[集合]]<br /> *[[集合論]]<br /> *[[集合の代数学]]<br /> <br /> {{集合論}}<br /> {{DEFAULTSORT:ふふんしゆうこう}}<br /> [[Category:集合論]]<br /> [[Category:初等数学]]<br /> [[Category:数学に関する記事]]<br /> <br /> [[ro:Mulțime#Submulțimi]]</div>

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