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Matematikan, bereziki multzo-teorian, azpimultzoa multzo bateko zenbait elementuz osatutako edozein multzoa da. In mathematics, set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B. A k-subset is a subset with k elements. Якщо X та Y — множини та будь-який елемент із X є також елементом із Y, то говорять, що: * X є підмножиною (частиною) Y, позначення — X ⊆ Y; * Y — надмножина (охоплююча множина) X, позначення — Y ⊇ X. Кожна множина Y є підмножиною себе самої.Підмножина Y, яка не збігається з Y називається точною підмножиною (або правильною чи власною частиною множини) Y.Якщо X — точна підмножина Y, то цей факт записується як X ⊂ Y.Відношення «бути підмножиною» має назву включення. 部分集合(ぶぶんしゅうごう)とは数学における概念の1つ。集合Aが集合Bの部分集合であるとは、AがBの一部の要素だけからなることである。AがB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包含関係があるという。 Die mathematischen Begriffe Teilmenge und Obermenge beschreiben eine Beziehung zwischen zwei Mengen. Ein anderes Wort für Teilmenge ist Untermenge. Für die mathematische Abbildung der Einbettung einer Teilmenge in ihre Grundmenge, die mathematische Funktion der Teilmengenbeziehung, wird die Inklusionsabbildung verwendet. ist eine Teilmenge von und ist eine Obermenge von , wenn jedes Element von auch in enthalten ist. Wenn zudem weitere Elemente enthält, die nicht in enthalten sind, so ist eine echte Teilmenge von und ist eine echte Obermenge von .Die Menge aller Teilmengen einer gegebenen Menge heißt die Potenzmenge von . 집합론에서 집합 B의 부분집합(部分集合, 영어: subset) A는, 모든 원소가 B에도 속하는 집합이다. 이런 관계를 주로 A ⊆ B라 표기한다. 예를 들어 집합 {1, 2}는 {1, 2, 3}의 부분집합이다. 벤 다이어그램에서는 부분집합 관계를 하나가 하나를 완전히 감싼 두 원으로 나타낸다. A = B인 경우에도 A는 B의 부분집합이 되는데, 그렇지 않은 부분집합을 진부분집합(眞部分集合, 영어: proper subset)이라고 한다. 임의의 집합의 원소에 일정한 제약을 가해 그 집합의 부분집합을 만들 수 있다. 이는 ZFC의 에도 반영된다. 집합의 모든 부분집합을 모아놓은 집합을 멱집합이라고 한다. En mathématiques, l’inclusion est une relation d'ordre entre ensembles. On dit qu'un ensemble A est inclus dans un ensemble B si tous les éléments de A sont aussi éléments de B. On dit dans ce cas que A est un sous-ensemble ou une partie de B, ou encore que B est sur-ensemble de A. Cette relation n'est pas symétrique a priori, car il peut y avoir des éléments du deuxième ensemble qui n'appartiennent pas au premier. Plus précisément, il y a inclusion dans les deux sens entre deux ensembles si et seulement si ces deux ensembles sont égaux. In matematica, e in particolare in teoria degli insiemi, l'inclusione, indicata con , è una relazione binaria tra insiemi definita nel seguente modo: "l'insieme è contenuto o incluso nell'insieme se, per ogni elemento , se appartiene a allora appartiene ad ". In simboli, dati due insiemi e , si ha: L'insieme si dice sottoinsieme di . Si parla, più propriamente, di inclusione stretta per indicare che ogni elemento di è anche elemento di ma che esistono elementi di che non sono elementi di . En matematiko, aparte en aroteorio, aro A estas subaro de aro B, se A estas "enhavata" ene de B. La interrilato de unu aro estante subaro de alia estas nomata kiel inkluziveco. Ĉiu aro estas subaro de si. Pli formale, Se A kaj B estas aroj kaj ĉiu de A estas ankaŭ ero de B, tiam: * A estas subaro de (aŭ estas inkluzivita en) B, skribata per A ⊆ B, aŭ ekvivalente * B estas superaro de (aŭ inkluziva) A, skribata per B ⊇ A. Por ĉiu aro S, inkluziveco estas rilato sur la aro de ĉiuj subaroj de S. في الرياضيات وبالتحديد في نظرية المجموعات، المجموعة الجزئية (بالإنجليزية: Subset)‏ مصطلح رياضي في فرع نظرية المجموعات. إذا كان كل عنصر في المجموعة A أيضاً عنصراً في المجموعة B تسمى عندها المجموعة A مجموعة جزئية من B. إذا كانت A مجموعة جزئية من B وB مجموعة جزئية من A، عندها يكون A = B. العلاقة بين مجموعة تكون مجموعة جزئية من مجموعة أخرى تسمى علاقة احتواء. Dalam matematika, terutama teori himpunan, suatu himpunan A adalah himpunan bagian atau subhimpunan dari himpunan B bila A "termuat" di dalam B. A dan B boleh jadi merupakan himpunan yang sama. Hubungan suatu himpunan yang menjadi himpunan bagian yang lain disebut sebagai "termasuk ke dalam" atau kadang-kadang "pemuatan". Himpunan B adalah superhimpunan dari A karena semua elemen A juga adalah elemen B. Ένα σύνολο X ονομάζεται υποσύνολο ενός συνόλου Y και συμβολίζουμε με , εάν κάθε στοιχείο του X είναι και στοιχείο (ανήκει) του Y δηλαδή ισχύει: Ακόμα χρησιμοποιούμε την ορολογία: το σύνολο X περιέχεται στο σύνολο Y ή ακόμα ότι το σύνολο Y είναι υπερσύνολο του συνόλου X και γράφουμε . Μπορούμε να θεωρήσουμε το ως τη σχέση που αποτελείται από όλα τα διατεταγμένα ζεύγη (X, Y) για τα οποία ισχύει . Παραδείγματα: * το σύνολο όλων των ανδρών είναι υποσύνολο του συνόλου όλων των ανθρώπων * * Αναφέρουμε ότι: το είναι υποσύνολο κάθε συνόλου και επίσης κάθε σύνολο Α είναι υποσύνολο του εαυτού του. En matemáticas, un conjunto B es subconjunto de otro conjunto A si todos los elementos de B pertenecen también a A. Decimos entonces que B «está contenido» dentro de A. Podzbiór – pewna „część” danego zbioru, czyli dla danego zbioru, nazywanego nadzbiorem, zbiór składający się z pewnej liczby jego elementów, np. żadnego, jednego, wszystkich. Pierwszy przypadek nazywa się podzbiorem pustym, drugi – podzbiorem jednoelementowym lub singletonem, trzeci – podzbiorem niewłaściwym. Подмно́жество в теории множеств — это понятие части множества. V matematice se jako podmnožina množiny A označuje taková množina B, o jejíchž všech prvcích platí, že jsou zároveň i prvky množiny A. Obdobně se může množina A označit jako nadmnožina množiny B. Tato fakta značíme , případně . Relace „být podmnožinou“ se nazývá také inkluze. Každá množina je svojí podmnožinou. Podmnožina množiny B, která jí není rovna, se označuje jako vlastní podmnožina množiny B. Tzn. žádná množina není svojí vlastní podmnožinou. Relace "být vlastní podmnožinou" se též nazývá ostrá inkluze. Un subconjunt és un conjunt format per elements d'un altre conjunt. Es diu que el primer conjunt és subconjunt del segon conjunt. Una manera més formal d'expressar això seria: Siguin X i Y dos conjunts, es diu que X és subconjunt de Y' quan tot element de X és també element de Y. Per exemple:A={1,2,3} i B={1,2,3,4,7}. Es pot dir que A és un subconjunt de B perquè tots els elements de A també pertanyen a B. La relació entre un subconjunt i un conjunt s'anomena inclusió i es representa pel símbol ⊆ o en la posició inversa ⊇. En l'exemple anterior, escriuríem A ⊆ B o B ⊇ A. In de verzamelingenleer is een deelverzameling van een gegeven verzameling een verzameling die geheel bevat is in (deel is van) de gegeven verzameling. Alle elementen van de deelverzameling zijn dus ook element van de gegeven verzameling. Als en verzamelingen zijn en ieder element van is ook een element van , dan is een deelverzameling van , genoteerd als: . Formeel: . Iedere verzameling is een deelverzameling van zichzelf, voor iedere verzameling geldt dus . De omgekeerde definitie is minder gebruikelijk. Uitgaande van dezelfde verzamelingen en zeggen we: omvat , genoteerd als . Em teoria dos conjuntos, quando todo elemento de um conjunto é também elemento de um conjunto dizemos que é um subconjunto ou uma parte de e denotamos (lê-se: está contido em ou é subconjunto de ou é uma parte de ) ou ainda (lê-se: contém ou é superconjunto de ou tem como parte). Esta relação é conhecida por inclusão de conjuntos. Em linguagem simbólica, Inom mängdteorin är en mängd A en delmängd av en mängd B om alla element som ingår i A även ingår i B. Detta skrivs A ⊆ B. Varje mängd är en delmängd av sig själv och den tomma mängden ∅ är en delmängd av alla mängder. Om A ⊆ B och B ⊆ A så följer A = B. Formellt definieras en delmängd som En delmängd uppfyller det formella sambandet En äkta delmängd A till en mängd B är en delmängd till B som inte är lika med B, det vill säga B innehåller element som inte finns i A. Ingen mängd är en äkta delmängd till sig själv och den tomma mängden är en äkta delmängd till alla icke-tomma mängder. 子集(subset)亦稱部分集合,爲某集合中一部分的集合;關係相反時則稱作父集、母集、超集。子集與父集关系上以“包含”稱呼。 若和为集合,且的所有元素都是的元素,则可表示為: * 是的子集(或称包含于); * 是的父集/超集(或称包含); 任何集合皆是本身的子集()。而的子集中不等于的集合,称为真子集,若是的真子集,写作。
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V matematice se jako podmnožina množiny A označuje taková množina B, o jejíchž všech prvcích platí, že jsou zároveň i prvky množiny A. Obdobně se může množina A označit jako nadmnožina množiny B. Tato fakta značíme , případně . Relace „být podmnožinou“ se nazývá také inkluze. Každá množina je svojí podmnožinou. Podmnožina množiny B, která jí není rovna, se označuje jako vlastní podmnožina množiny B. Tzn. žádná množina není svojí vlastní podmnožinou. Relace "být vlastní podmnožinou" se též nazývá ostrá inkluze. Die mathematischen Begriffe Teilmenge und Obermenge beschreiben eine Beziehung zwischen zwei Mengen. Ein anderes Wort für Teilmenge ist Untermenge. Für die mathematische Abbildung der Einbettung einer Teilmenge in ihre Grundmenge, die mathematische Funktion der Teilmengenbeziehung, wird die Inklusionsabbildung verwendet. ist eine Teilmenge von und ist eine Obermenge von , wenn jedes Element von auch in enthalten ist. Wenn zudem weitere Elemente enthält, die nicht in enthalten sind, so ist eine echte Teilmenge von und ist eine echte Obermenge von .Die Menge aller Teilmengen einer gegebenen Menge heißt die Potenzmenge von . Den Begriff Teilmenge prägte Georg Cantor – der „Erfinder“ der Mengenlehre – ab 1884; das Symbol der Teilmengenrelation wurde von Ernst Schröder 1890 in seiner „Algebra der Logik“ eingeführt. Якщо X та Y — множини та будь-який елемент із X є також елементом із Y, то говорять, що: * X є підмножиною (частиною) Y, позначення — X ⊆ Y; * Y — надмножина (охоплююча множина) X, позначення — Y ⊇ X. Кожна множина Y є підмножиною себе самої.Підмножина Y, яка не збігається з Y називається точною підмножиною (або правильною чи власною частиною множини) Y.Якщо X — точна підмножина Y, то цей факт записується як X ⊂ Y.Відношення «бути підмножиною» має назву включення. Ένα σύνολο X ονομάζεται υποσύνολο ενός συνόλου Y και συμβολίζουμε με , εάν κάθε στοιχείο του X είναι και στοιχείο (ανήκει) του Y δηλαδή ισχύει: Ακόμα χρησιμοποιούμε την ορολογία: το σύνολο X περιέχεται στο σύνολο Y ή ακόμα ότι το σύνολο Y είναι υπερσύνολο του συνόλου X και γράφουμε . Μπορούμε να θεωρήσουμε το ως τη σχέση που αποτελείται από όλα τα διατεταγμένα ζεύγη (X, Y) για τα οποία ισχύει . Παραδείγματα: * το σύνολο όλων των ανδρών είναι υποσύνολο του συνόλου όλων των ανθρώπων * * Αναφέρουμε ότι: το είναι υποσύνολο κάθε συνόλου και επίσης κάθε σύνολο Α είναι υποσύνολο του εαυτού του. * για κάθε σύνολο Α * για κάθε σύνολο Α 部分集合(ぶぶんしゅうごう)とは数学における概念の1つ。集合Aが集合Bの部分集合であるとは、AがBの一部の要素だけからなることである。AがB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包含関係があるという。 Un subconjunt és un conjunt format per elements d'un altre conjunt. Es diu que el primer conjunt és subconjunt del segon conjunt. Una manera més formal d'expressar això seria: Siguin X i Y dos conjunts, es diu que X és subconjunt de Y' quan tot element de X és també element de Y. Per exemple:A={1,2,3} i B={1,2,3,4,7}. Es pot dir que A és un subconjunt de B perquè tots els elements de A també pertanyen a B. La relació entre un subconjunt i un conjunt s'anomena inclusió i es representa pel símbol ⊆ o en la posició inversa ⊇. En l'exemple anterior, escriuríem A ⊆ B o B ⊇ A. Seguint la definició, tot conjunt A és subconjunt d'ell mateix. Per això, es parla de subconjunts propis d'A per a referir-se als subconjunts d'A que no són ell mateix. 子集(subset)亦稱部分集合,爲某集合中一部分的集合;關係相反時則稱作父集、母集、超集。子集與父集关系上以“包含”稱呼。 若和为集合,且的所有元素都是的元素,则可表示為: * 是的子集(或称包含于); * 是的父集/超集(或称包含); 任何集合皆是本身的子集()。而的子集中不等于的集合,称为真子集,若是的真子集,写作。 Подмно́жество в теории множеств — это понятие части множества. 집합론에서 집합 B의 부분집합(部分集合, 영어: subset) A는, 모든 원소가 B에도 속하는 집합이다. 이런 관계를 주로 A ⊆ B라 표기한다. 예를 들어 집합 {1, 2}는 {1, 2, 3}의 부분집합이다. 벤 다이어그램에서는 부분집합 관계를 하나가 하나를 완전히 감싼 두 원으로 나타낸다. A = B인 경우에도 A는 B의 부분집합이 되는데, 그렇지 않은 부분집합을 진부분집합(眞部分集合, 영어: proper subset)이라고 한다. 임의의 집합의 원소에 일정한 제약을 가해 그 집합의 부분집합을 만들 수 있다. 이는 ZFC의 에도 반영된다. 집합의 모든 부분집합을 모아놓은 집합을 멱집합이라고 한다. Matematikan, bereziki multzo-teorian, azpimultzoa multzo bateko zenbait elementuz osatutako edozein multzoa da. في الرياضيات وبالتحديد في نظرية المجموعات، المجموعة الجزئية (بالإنجليزية: Subset)‏ مصطلح رياضي في فرع نظرية المجموعات. إذا كان كل عنصر في المجموعة A أيضاً عنصراً في المجموعة B تسمى عندها المجموعة A مجموعة جزئية من B. إذا كانت A مجموعة جزئية من B وB مجموعة جزئية من A، عندها يكون A = B. العلاقة بين مجموعة تكون مجموعة جزئية من مجموعة أخرى تسمى علاقة احتواء. En matematiko, aparte en aroteorio, aro A estas subaro de aro B, se A estas "enhavata" ene de B. La interrilato de unu aro estante subaro de alia estas nomata kiel inkluziveco. Ĉiu aro estas subaro de si. Pli formale, Se A kaj B estas aroj kaj ĉiu de A estas ankaŭ ero de B, tiam: * A estas subaro de (aŭ estas inkluzivita en) B, skribata per A ⊆ B, aŭ ekvivalente * B estas superaro de (aŭ inkluziva) A, skribata per B ⊇ A. Se A estas subaro de B, sed A estas ne egala al B, tiam A estas ankaŭ strikta (aŭ pozitiva) subaro de B. Ĉi tio estas skribita kiel A ⊂ B. En la sama vojo, B ⊃ A signifas ke B estas strikta superaro de A. Simboloj ⊆ kaj ⊂ estas analoga al ≤ kaj <. Ekzemple, se A estas (larĝsenca) subaro de B (skribita kiel A ⊆ B), tiam la kvanto da eroj en A estas malpli ol aŭ egala al la kvanto da eroj en B (skribita kiel |A| ≤ |B|). Ankaŭ, por aroj A kaj B, se A ⊂ B tiam |A| < |B|. Por ĉiu aro S, inkluziveco estas rilato sur la aro de ĉiuj subaroj de S. Inom mängdteorin är en mängd A en delmängd av en mängd B om alla element som ingår i A även ingår i B. Detta skrivs A ⊆ B. Varje mängd är en delmängd av sig själv och den tomma mängden ∅ är en delmängd av alla mängder. Om A ⊆ B och B ⊆ A så följer A = B. Formellt definieras en delmängd som En delmängd uppfyller det formella sambandet En äkta delmängd A till en mängd B är en delmängd till B som inte är lika med B, det vill säga B innehåller element som inte finns i A. Ingen mängd är en äkta delmängd till sig själv och den tomma mängden är en äkta delmängd till alla icke-tomma mängder. Om A är en delmängd till B sägs B vara en övermängd till A, vilket betecknas (A är en äkta delmängd av B om och endast om B är en äkta övermängd till A). In de verzamelingenleer is een deelverzameling van een gegeven verzameling een verzameling die geheel bevat is in (deel is van) de gegeven verzameling. Alle elementen van de deelverzameling zijn dus ook element van de gegeven verzameling. Als en verzamelingen zijn en ieder element van is ook een element van , dan is een deelverzameling van , genoteerd als: . Formeel: . Iedere verzameling is een deelverzameling van zichzelf, voor iedere verzameling geldt dus . De omgekeerde definitie is minder gebruikelijk. Uitgaande van dezelfde verzamelingen en zeggen we: omvat , genoteerd als . En mathématiques, l’inclusion est une relation d'ordre entre ensembles. On dit qu'un ensemble A est inclus dans un ensemble B si tous les éléments de A sont aussi éléments de B. On dit dans ce cas que A est un sous-ensemble ou une partie de B, ou encore que B est sur-ensemble de A. Cette relation n'est pas symétrique a priori, car il peut y avoir des éléments du deuxième ensemble qui n'appartiennent pas au premier. Plus précisément, il y a inclusion dans les deux sens entre deux ensembles si et seulement si ces deux ensembles sont égaux. L'inclusion se note majoritairement avec le symbole « ⊂ » introduit par Schröder, même si d'autres auteurs réservent ce symbole à l'inclusion stricte (c'est-à-dire excluant le cas d'égalité), suivant ainsi la norme ISO. L'inclusion au sens large peut alors être notée avec le symbole « ⊆ » de Felix Hausdorff, par analogie avec les symboles de comparaison numériques. Pour lever l'ambiguïté, l'inclusion stricte peut aussi être notée « ⊊ », à ne pas confondre avec la négation de l'inclusion, qui se note « ⊄ » ou « ⊈ ». Tous ces symboles peuvent être réfléchis pour représenter les relations réciproques. In matematica, e in particolare in teoria degli insiemi, l'inclusione, indicata con , è una relazione binaria tra insiemi definita nel seguente modo: "l'insieme è contenuto o incluso nell'insieme se, per ogni elemento , se appartiene a allora appartiene ad ". In simboli, dati due insiemi e , si ha: L'insieme si dice sottoinsieme di . Si parla, più propriamente, di inclusione stretta per indicare che ogni elemento di è anche elemento di ma che esistono elementi di che non sono elementi di . Nel caso in cui tutti gli elementi di appartengono anche a si parla di sottoinsieme improprio (in altre parole ogni insieme è un sottoinsieme improprio di sé stesso). Si parla di sottoinsieme proprio se almeno un elemento di non è compreso nell'insieme , cioè nel caso dell'inclusione stretta. Il simbolo usato per indicare un sottoinsieme è , mentre il simbolo per indicare un sottoinsieme proprio è . Tuttavia spesso viene usata una notazione alternativa che indica con un sottoinsieme e con un sottoinsieme proprio (quest'ultima si usa anche quando si vuole mettere in evidenza che non coincide con ). Analogamente si definisce il concetto di sovrainsieme; il simbolo usato è (oppure ) per il sovrainsieme, e (oppure ) per il sovrainsieme proprio. Dalam matematika, terutama teori himpunan, suatu himpunan A adalah himpunan bagian atau subhimpunan dari himpunan B bila A "termuat" di dalam B. A dan B boleh jadi merupakan himpunan yang sama. Hubungan suatu himpunan yang menjadi himpunan bagian yang lain disebut sebagai "termasuk ke dalam" atau kadang-kadang "pemuatan". Himpunan B adalah superhimpunan dari A karena semua elemen A juga adalah elemen B. Podzbiór – pewna „część” danego zbioru, czyli dla danego zbioru, nazywanego nadzbiorem, zbiór składający się z pewnej liczby jego elementów, np. żadnego, jednego, wszystkich. Pierwszy przypadek nazywa się podzbiorem pustym, drugi – podzbiorem jednoelementowym lub singletonem, trzeci – podzbiorem niewłaściwym. En matemáticas, un conjunto B es subconjunto de otro conjunto A si todos los elementos de B pertenecen también a A. Decimos entonces que B «está contenido» dentro de A. Em teoria dos conjuntos, quando todo elemento de um conjunto é também elemento de um conjunto dizemos que é um subconjunto ou uma parte de e denotamos (lê-se: está contido em ou é subconjunto de ou é uma parte de ) ou ainda (lê-se: contém ou é superconjunto de ou tem como parte). Esta relação é conhecida por inclusão de conjuntos. Em linguagem simbólica, In mathematics, set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B. A k-subset is a subset with k elements. The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation.
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