one

Bahamian Creole: wan

Belizean Creole: wan

Bislama: wan

Gullah: one

Jamaican Creole: wan

Nigerian Pidgin: wọ́n, wọn

Sranan Tongo: wan

Tok Pisin: wan

→ Atong (India): wan

→ Fanagalo: wan

PIE word
 *h₁óynos
From Middle English oon, on, oan, an, from Old English ān (“one”), from Proto-West Germanic *ain, from Proto-Germanic *ainaz (“one”), from Proto-Indo-European *h₁óynos (“single, one”).
Cognate with Scots ae, ane, wan, yin (“one”); North Frisian ån (“one”); Saterland Frisian aan (“one”); West Frisian ien (“one”); Dutch een, één (“one”); German Low German een; German ein, eins (“one”); Danish en (“one”); Swedish en (“one”); Norwegian Nynorsk ein (“one”), Icelandic einn (“one”); Latin ūnus (“one”) (Old Latin oinos); Russian оди́н (odín), Spanish uno. Doublet of a, an, and Uno.
The use as an indefinite personal pronoun may have been influenced by unrelated French on, although the Germanic languages widely use cognates for the same sense (usually in non-subject function, but also in subject function, e.g. Luxembourgish een).
Verb form from Middle English onen.

inh

wone (alternative obsolete)
I (alternative)

one's damned if one does and one's damned if one doesn't

(set theory) The cardinality of the smallest nonempty set.


transitive

From Latin trānsitīvus, from trānsitus, from trāns (“across”) + itus, from eō (“to go”).

"Is an ancestor of" is a transitive relation: if Alice is an ancestor of Bob, and Bob is an ancestor of Carol, then Alice is an ancestor of Carol.

(set theory, of a relation on a set) Having the property that if an element a is related to b and b is related to c, then a is necessarily related to c.


singular

en-adj

singular (comparative more singular, superlative most singular)

more singular (comparative)
most singular (superlative)
sing. (alternative abbreviation)
sg. (alternative abbreviation)
sg (alternative abbreviation)

From Middle English singuler, from Old French, from Latin singulāris (“alone of its kind”), from Latin singulus (“single”).

(set theory, of a cardinal number) Not equal to its own cofinality.


mouse

en-noun

mouse (plural mice or (computing) mouses)

mice (plural)
mouses (plural)
mowse (alternative obsolete)

→ Bengali: মাউস (maus)

→ Brazilian Portuguese: mouse

→ French: souris (semantic loan)

→ Georgian: მაუსი (mausi)

→ Italian: mouse

→ Japanese: マウス (mausu)

→ Korean: 마우스 (mauseu)

→ Romanian: mouse

→ Spanish: mouse

desc

→ Thai: เมาส์ (máo), เม้าส์ (máo)

→ Thai: เมาส์ (máo), เม้าส์ (máo)

→ Vietnamese: chuột (semantic loan)

From Middle English mous, from Old English mūs, from Proto-West Germanic *mūs, from Proto-Germanic *mūs, from Proto-Indo-European *múHs.
Cognates
Germanic cognates include Old Frisian mūs, Old Saxon mūs (German Low German Muus), Dutch muis, Old High German mūs (German Maus), Old Norse mús (Swedish mus, Danish mus, Norwegian mus, Icelandic mús, Faroese mús).
Indo-European cognates include Ancient Greek μῦς (mûs), Latin mūs, Spanish mur, Armenian մուկ (muk), Old Church Slavonic мꙑшь (myšĭ) (Russian мышь (myšʹ)), Albanian mi, Persian موش (muš), Northern Kurdish mişk, Sanskrit मूष् (mūṣ).
The computing sense was coined by American engineer Bill English in 1965 and first used publicly in a publication titled "Computer-Aided Display Control", in reference to the similarity with the animal.

coin

the best laid plans of mice and men go oft astray

the best laid plans of mice and men often go awry

(set theory) A small model of (a fragment of) Zermelo-Fraenkel set theory with desirable properties (depending on the context).


ordinal number

ординал

generalized kind of number to denote the length of a well-order on a set

1950, Frederick Bagemihl (translator), Erich Kamke, Theory of Sets, Dover (Dover Phoenix), 2006, page 137,
For not only do the antinomies a) to e) disappear when we admit as elements of sets only such sets, ordinal numbers, and cardinal numbers as are bounded above by a fixed cardinal number, but we see also that paradoxes always arise if we collect into a set any sets, cardinal numbers, or ordinal numbers which are not bounded above by a fixed cardinal number.

1960 [D. Van Nostrand], Paul R. Halmos, Naive Set Theory, 2017, Dover, Republication, page 80,
Is there a set that consists exactly of all the ordinal numbers? It is easy to see that the answer must be no. If there were such a set, then we could form the supremum of all ordinal numbers. That supremum would be an ordinal number greater than or equal to every ordinal number. Since, however, for each ordinal number there exists a strictly greater one (for example, its successor), this is impossible; it makes no sense to speak of the "set" of all ordinals.

If #92;alpha is an ordinal number, then by definition any two well-ordered sets of type #92;alpha are similar, i.e., there exists a one-to-one mapping from one set to the other. Consequently these sets have the same cardinality. Consequently to any ordinal number #92;alpha we may assign a cardinal number, the common cardinality of all well-ordered sets of type #92;alpha.

2009, Marek Kuczma, edited by Attila Gilányi, An Introduction to the Theory of Functional Equations and Inequalities, 2nd edition, Springer (Birkhäuser), page 10:

equivalence relation
equivalence relation

(set theory) Such a number generalised to correspond to any cardinal number (the size of some set); formally, the order type of some well-ordered set of some cardinality a, which represents an equivalence class of well-ordered sets (exactly those of cardinality a) under the equivalence relation "existence of an order-preserving bijection".


chain

From Middle English cheyne, chaine, from Old French chaine, chaene (“chain”), from Latin catēna (“chain”), from Proto-Indo-European *kat- (“to braid, twist; hut, shed”). Doublet of catena. Displaced native Middle English rakil and rakent (from Old English racente (“chain”)); see rackan.

der

a chain is only as strong as its weakest link

We first find an approximation of the chain partition, i.e. a small but not minimum size set of chains which cover all elements of the poset.

2003, Jeremy P. Spinrad, Efficient Graph Representations, American Mathematical Society, page 108:

(mathematics, set theory, order theory) A totally ordered set, especially a totally ordered subset of a poset.


AC

ACs (plural)
a.c. (alternative)
A/C (alternative)
a/c (alternative)
A.C. (alternative)
a.c. (alternative)
ac (alternative)
A.C. (alternative)
A.C. (alternative)
A. C. (alternative)
a.c. (alternative)

(set theory) Initialism of axiom of choice.


CH

en-noun

CH (countable and uncountable, plural CHs)

continuum hypothesis
continuum hypothesis#English

(set theory) Initialism of continuum hypothesis.


family

family (countable and uncountable, plural families)

Jamaican Creole: faambli, fambili

Tok Pisin: famili

Sranan Tongo: famiri

→ Chuukese: famini

→ Malay: famili

→ Maori: whāmere

From Late Middle English famylye, from Latin familia (“a household”). Displaced native Old English hīred. Doublet of familia.

عَائِلَة

group of people related by blood, marriage, law, or custom

the family that lays together stays together

the family that prays together stays together

the family that sleeps together keeps together

Let #92;mathcalF be a family of subsets over S.

Any group or aggregation of things classed together as kindred or related from possessing in common characteristics which distinguish them from other things of the same order.

(set theory, countable) A collection of sets, especially of subsets of a given set.


ideal

From French idéal, from Late Latin ideālis (“existing in idea”), by surface analysis, idea + -al, from Latin idea (“idea”); see idea.
In mathematics, the noun ring theory sense was first introduced by German mathematician Richard Dedekind in his 1871 edition of a text on number theory. The concept was quickly expanded to ring theory and later generalised to order theory. The set theory and Lie theory senses can be regarded as applications of the order theory sense.

Formally, an ideal I of a given set X is a nonempty subset of the powerset #92;mathcal#123;P#125;(X) such that: (1)#92;#92;emptyset#92;inI, (2)#92;A#92;inI#92;andB#92;subseteqA#92;impliesB#92;inI and (3)#92;A,B#92;inI#92;impliesA#92;cupB#92;inI.

(set theory) A collection of sets, considered small or negligible, such that every subset of each member and the union of any two members are also members of the collection.


set

From Middle English set, sete, sette (“that which is set, the act of setting, seat”), from Old English set (“setting, seat, a place where people remain, habitation, camp, entrenchment, a place where animals are kept, stall, fold”) and Old English seten (“a set, shoot, slip, branch; a nursery, plantation; that which is planted or set; a cultivated place; planting, cultivation; a setting, putting; a stopping; occupied land”), related to Old English settan (“to set”). Compare Middle Low German gesette (“a set, suite”), Old English gesetl (“assembly”). According to Skeat, in senses denoting a group of things or persons, representing an alteration of sept, from Old French sette (“a religious sect”), from Medieval Latin secta (“retinue”), from Latin secta (“a faction”). See sect. It is quite possible that the modern word is more of a merger between both, however.

cog

(set theory) A collection of zero or more objects, possibly infinite in size, and disregarding any order or repetition of the objects which may be contained within it.


ball

head

ball m (genitive singular baill, nominative plural baill)

baill (genitive singular)
baill (nominative plural)
ball (indefinite nominative singular)
baill (indefinite nominative plural)
a bhaill (indefinite singular vocative)
a bhalla (indefinite plural vocative)
baill (genitive indefinite singular)
ball (genitive indefinite plural)
ball (dative indefinite singular)
baill (dative indefinite plural)
an ball (definite nominative singular)
na baill (definite nominative plural)
an bhaill (definite genitive singular)
na mball (definite genitive plural)
leis an mball (dative definite singular)
don bhall (dative definite singular)
leis na baill (dative definite plural)
ball
bhall
mball

From Old Irish ball, from Proto-Celtic *ballos, from Proto-Indo-European *bʰel- (“to blow, swell, inflate”); compare English ball, Greek φαλλός (phallós, “penis”).


power

en-noun

power (countable and uncountable, plural powers)

powers (plural)
powre (alternative obsolete)

→ German: Power

/ˈpʌʊ̯.(ə)ɹ/

From Middle English power, poer, from Old French poeir, from Vulgar Latin potēre, from Latin posse, whence English potent. Compare French pouvoir. Displaced the native Old English anweald.

հզորություն

physics: measure of the rate of doing work or transferring energy


domain

range

antonym(s) of “domain of definition of a function”

From Middle English demayne, demain (“rule”), from Old French demeine, demaine, demeigne, domaine (“power”), (French domaine), from Latin dominium (“property, right of ownership”), from dominus (“master, proprietor, owner”). See dame, and compare demain, danger, dungeon. Doublet of demesne, domino.

definisieversameling

(mathematics) the set on which a function is defined

(mathematics, set theory) The set of input (argument) values for which a function is defined.

(more generally, of a binary relation R between A and B) The set A; The subset of A consisting of elements a of A such that there exists an element b in B with (a,b) in R.


勢

Phono-semantic compound (形聲 /形声, OC *hŋjeds): phonetic 埶 (OC *ŋeds) + semantic 力 (“power”).
Nominalisation of 設 (OC *ŋ̊et, “to set up”), i.e. “that which is set up” (Schuessler, 2007; Baxter and Sagart, 2014). Compare the following quote from Han Feizi

Han compound

The setup (*ŋ̊et-s) of which I am speaking refers to what is set up (*ŋ̊et) by men.

Wú suǒ wéi yán shì zhě, yán rén zhī suǒ shè yě.

吾所為言勢者，言人之所設也。

吾所为言势者，言人之所设也。


kernel

→ Japanese: カーネル (kāneru)

From Middle English kernel, kirnel, kürnel, from Old English cyrnel, from Proto-West Germanic *kurnil, diminutive of Proto-Germanic *kurną (“seed, grain, corn”), equivalent to corn + -le. Cognate with Yiddish קערנדל (kerndl), Middle Dutch kernel, cornel, Middle High German kornel. Related also to Old Norse kjarni (“kernel”).

ядро

(mathematics, algebra) set of elements mapped to zero

If a function is continuous then its kernel is a closed set.

(mathematics, set theory, of a function f) The set of pairs of elements in the domain of f which are mapped to the same value; the equivalence relation x≡y⟺f(x)=f(y).


order

en-noun

order (countable and uncountable, plural orders)

orders (plural)
ordre (alternative obsolete)

→ Swahili: oda

From Middle English ordre, from Old French ordre, ordne, ordene (“order, rank”), from Latin ōrdinem, accusative of ōrdō (“row, rank, regular arrangement”, literally “row of threads in a loom”), from Proto-Italic *ordō (“to arrange”), probably ultimately from Proto-Indo-European *h₂or-d-, from *h₂er-.
Related to Latin ōrdior (“begin”, literally “begin to weave”). In sense “request for purchase”, compare bespoke. Doublet of ordo.

cog

1911 [Cambridge University Press], William Burnside, Theory of Groups of Finite Order, 2nd Edition, Reprint, Dover (Dover Phoenix), 2004, page 222,
In this case, the conjugate set contains n(n − 1)/x(x − 1) distinct sub-groups of order m, and H is therefore self-conjugate in a group K of order x(x − l)m.

For various reasons it turns out to be better to enlarge this set of invariants to include suitable normalizers of subgroups of odd prime order.

2000, Michael Aschbacher, Finite Group Theory, 2nd edition, Cambridge University Press, page 260:

(set theory, of a set or algebraic structure) The number of elements contained within (the given object); formally, the cardinality (of the given object).


class

en-noun

class (countable and uncountable, plural classes)

→ Assamese: ক্লাছ (klas)

desc

→ Bengali: ক্লাস (klaś), কেলাস (kelaś) — nonstandard

q

or from French <i class="Latn mention" lang="fr">classe</i>

→ Gulf Arabic: كلاس (klās) (or from French classe)

→ Hindustani:

Hindi: क्लास (klās)

Urdu: کلاس

→ Japanese: クラス (kurasu)

→ Korean: 클래스 (keullaeseu)

→ Nepali: क्लास (klās)

→ Scottish Gaelic: clas

→ Thai: คลาส (kláas)

→ Yoruba: kíláàsì

From Middle French classe, from Latin classis (“a class or division of the people, assembly of people, the whole body of citizens called to arms, the army, the fleet, later a class or division in general”), from Proto-Indo-European *kelh₁- (“to call, shout”). Doublet of clas and classis.

class action

class-action, class action lawsuit, class suit

The class of all sets is not a set.

Every set is a class, but classes are not generally sets. A class that is not a set is called a proper class.

In the present section we shall discuss the various systems of set theory which admit, beside sets, also classes. Classes are like sets, except that they can be very comprehensive; an extreme example of a class is the class which contains all sets.[…]The main point which will, in our opinion, emerge from this analysis is that set theory with classes and set theory with sets only are not two separate theories; they are, essentially, different formulations of the same underlying theory.

1973, Abraham Fraenkel, Yehoshua Bar-Hillel, Azriel Lévy, Foundations of Set Theory, 2nd edition, Elsevier, page 119:

(set theory) A collection of sets definable by a shared property, especially one which is not itself a set (in which case the class is called proper).

դաս

set theory: collection of sets definable by a shared property


atom

→ Swahili: atomi

From Middle English attome, from Middle French athome, from Latin atomus (“smallest particle”), from Ancient Greek ἄτομος (átomos, “indivisible”), from ἀ- (a-, “not”) + τέμνω (témnō, “I cut”). Atoms are so named because historically the notion was that they were indivisible, given that chemically they are indeed so; the splitting of atoms awaited a later era of science and technology, and nonchemical means. Doublet of atomus.

(mathematics, set theory) An element of a set that is not itself a set; an urelement.


類

Phono-semantic compound (形聲 /形声, OC *ruːls, *ruds): phonetic 頪 (OC *roːds) + semantic 犬.
From Proto-Sino-Tibetan *rus (“bone”) (STEDT).

Han compound

Yóu suǒyǒu jíhé gòuchéng de lèi bùshì jíhé.

由所有集合構成的類不是集合。

由所有集合构成的类不是集合。


omega

From Middle English, from Ancient Greek ὦ μέγα (ô méga), meaning “great ω” (omega is a long vowel in Ancient Greek).

(set theory) A transfinite ordinal number referring to the next position after ordering a countably infinite set.


homogeneous

From Medieval Latin homogeneus, from Ancient Greek ὁμογενής (homogenḗs, “of the same race, family or kind”), from ὁμός (homós, “same”) + γένος (génos, “kind”). Compare homo- (“same”) and -ous (adjectival suffix).

der

(mathematics) In any of several technical senses uniform; scalable; having its behavior or form determined by, or the same as, its behavior on or form at a smaller component (of its domain of definition, of itself, etc.).

(set theory, order theory, of a relation) Holding between a set and itself; being an endorelation.


intersection

/ˈɪntəɹˌsɛkʃən/

/ˌɪntəɹˈsɛkʃən/

From Middle French intersection and its etymon Latin intersectiō. By surface analysis, intersect + -ion.

(set theory) The set containing all the elements that are common to two or more sets.


algebra

en-noun

algebra (countable and uncountable, plural algebras)

⇒ Chinese: 代數 /代数 (dàishù)

→ Japanese: 代数

→ Korean: 대수 (daesu)

→ Vietnamese: đại số

/ˈæl.d͡ʒɪ.bɹə/

/ˈæl.d͡ʒə.bɹə/

Borrowed from Medieval Latin algebra, from the Arabic word الْجَبْر (al-jabr, “reunion, resetting of broken parts”) in the title of al-Khwarizmi's influential work الْكِتَاب الْمُخْتَصَر فِي حِسَاب الْجَبْر وَالْمُقَابَلَة (al-kitāb al-muḵtaṣar fī ḥisāb al-jabr wa-l-muqābala, “The Compendious Book on Calculation by Completion and Balancing”).

mathematical analysis
mathematical analysis

(countable, set theory, mathematical analysis) A collection of subsets of a given set, such that this collection contains the empty set, and the collection is closed under unions and complements (and thereby also under intersections and differences).


union

union (countable and uncountable, plural unions)

→ Bengali: ইউনিয়ন (iuniẏon) (learned)

From Middle English unyoun, from Old French union, from Late Latin ūniō, ūniōnem (“oneness, unity”), from Latin ūnus (“one”). Doublet of unio.

der

(countable, set theory) The set containing all of the elements of two or more sets.


union

en-verb

union (third-person singular simple present unions, present participle unioning, simple past and past participle unioned)

unions (present singular third-person)
unioning (participle present)
unioned (participle past)
unioned (past)

(set theory) To combine sets using the union operation.


element

From Middle English element, from Old French element, from Latin elementum (“a first principle, element, rudiment”) (see further etymology there).
The verb is from Middle English elementen, from the noun.

element

simplest or essential part or principle of anything

[…] the majority of names being compounded of two elements chosen from a stock of special name-words.

1945, E[lizabeth] G[idley] Withycombe, “Introduction”, in The Oxford Dictionary of English Christian Names, Oxford, Oxfordshire: Clarendon Press, →OCLC, page xi:

One of the simplest or essential parts or principles of which anything consists, or upon which the constitution or fundamental powers of anything are based.

(set theory) One of the objects in a set.


element

head

element n (singular definite elementet, plural indefinite elementer)

elementet (definite singular)
elementer (indefinite plural)
element (indefinite nominative singular)
elementet (definite nominative singular)
elementer (indefinite nominative plural)
elementerne (definite nominative plural)
elements (genitive indefinite singular)
elementets (definite genitive singular)
elementers (genitive indefinite plural)
elementernes (definite genitive plural)


element

nl-noun

element n (plural elementen, diminutive elementje n)

elementen (plural)
elementje (diminutive neuter)

→ Indonesian: elemen

From Middle Dutch element, from Old French element, from Latin elementum (“a first principle, element, rudiment”), of uncertain origin (see further etymology there).


amorphous

en-adj

amorphous (comparative more amorphous, superlative most amorphous)

more amorphous (comparative)
most amorphous (superlative)

From Ancient Greek ἄμορφος (ámorphos, “without form, shapeless, deformed”) (itself from ἀ- (a-, “without”) + μορφή (morphḗ, “form”) + -ous. By surface analysis, a- + -morphous.

der

(set theory, of a set) Infinite and not the disjoint union of two infinite subsets.


proper

en-adj

proper (comparative more proper, superlative most proper)

more proper (comparative)
most proper (superlative)
propa (alternative Geordie)
propre (alternative obsolete)

From Middle English propre, from Anglo-Norman proper, propre, Old French propre (French: propre), from Latin proprius.

luonnonvärinen

heraldry: having its natural or usual coloration

(set theory, of a class) Not being a set.

Accurate, strictly applied.
Not being a set.


choice

en-noun

choice (countable and uncountable, plural choices)

choices (plural)
choise (alternative obsolete)
choyse (alternative obsolete)

From Middle English chois, from Old French chois (“choice”), from choisir (“to choose, perceive”), possibly via assumed Vulgar Latin *causīre (“to choose”), from Gothic 𐌺𐌰𐌿𐍃𐌾𐌰𐌽 (kausjan, “to make a choice, taste, test, choose”), from Proto-Germanic *kauzijaną, from *keusaną (“to choose”), from Proto-Indo-European *ǵews- (“to choose”). Akin to Old High German kiosan (“to choose”), Old English ċēosan (“to choose”), Old Norse kjósa (“to choose”). More at choose.

der

you pays your money and you takes your choice

you pays your nickel and you takes your choice

5. ZF* is the theory obtained from the aforementioned axiomatics (without choice) by adding the Axiom of Inaccessible Cardinals to be explained in the next secion; similarly, we get ZFC*.

2016 July 15, Decio Krause, Jonas R.B. Arenhart, The Logical Foundations of Scientific Theories: Languages, Structures, and Models (Routledge Studies in the Philosophy of Mathematics and Physics), Routledge, →ISBN, page 51:

(set theory) Ellipsis of axiom of choice.


asymmetric

en-adj

asymmetric (comparative more asymmetric, superlative most asymmetric)

more asymmetric (comparative)
most asymmetric (superlative)

From a- + symmetric, as Ancient Greek ἀσυμμετρία (asummetría, “disproportion, deformity”).

der

(set theory) Of a relation R on a set S: having the property that for any two elements of S (not necessarily distinct), at least one is not related to the other via R.


symmetric

en-adj

symmetric (comparative more symmetric, superlative most symmetric)

more symmetric (comparative)
most symmetric (superlative)

From symmetry + -ic, from Latin symmetria, from Ancient Greek συμμετρία (summetría).

"Is a sibling of" is a symmetric relation.

(set theory) Of a relation R on a set S, such that xRy if and only if yRx for all members x and y of S (that is, if the relation holds between any element and a second, it also holds between the second and the first).

symétrique

of a relation R on a set S, such that xRy if and only if yRx for all members x and y of S


member

→ Japanese: メンバー (menbā)

→ Korean: 멤버 (membeo)

→ Malay: member

→ Punjabi: ਮੈਂਬਰ (maimbar)

From Middle English membre, from Old French membre, from Latin membrum (“limb, body part”), from Proto-Italic *memzrom, from Proto-Indo-European *mḗms, *mēms-rom (“flesh”). Akin to Gothic 𐌼𐌹𐌼𐌶 (mimz, “meat, flesh”), Crimean Gothic menus.
Coexists with native Middle English lim, limb (“member, limb, joint”) (from Old English lim (“limb, joint, main branch”)), and displaced Middle English lith (“limb, joint, member”) (from Old English liþ (“limb, member, join, tip”)).


infinite

en-adj

infinite (comparative more infinite, superlative most infinite)

more infinite (comparative)
most infinite (superlative)

From Middle English infinite, from Old French infinit and its etymon Latin īnfīnītus, from in- (“not”) + fīnis (“end”) + the perfect passive participle ending -ītus. By surface analysis, in- + finite. Doublet of infinito. Displaced native Old English unġeendodlīċ.

For any infinite set, there is a 1-1 correspondence between it and at least one of its proper subsets. For example, there is a 1-1 correspondence between the set of natural numbers and the set of squares of natural numbers, which is a proper subset of the set of natural numbers.

2009, Brandon C. Look, “Symbolic Logic II, Lecture 2: Set Theory”, in www.uky.edu/~look, archived from the original on 2018-06-19:

(set theory, of a set) Having infinitely many elements.


complement

en-noun

complement (countable and uncountable, plural complements)

From Middle English complement, from Latin complēmentum (“that which fills up or completes”), from compleō (“I fill up, I complete”) (English complete). Doublet of compliment. The verb is from the noun.

der

The complement of the odd numbers is the even numbers, relative to the natural numbers.

(set theory) Given two sets, the set containing one set's elements that are not members of the other set (whether a relative complement or an absolute complement).


keluarga

kêluarga (canonical)
keluarga-keluarga (plural)

From Malay keluarga (“family”). Compare to Javanese ꦏꦸꦭꦮꦂꦒ (kulawarga) and ꦏꦮꦸꦭꦮꦂꦒ (kawulawarga).

(set theory) class (a collection of sets definable by a shared property)


bijection

PIE word
 *dwóh₁
From French bijection, introduced by Nicolas Bourbaki in their treatise Éléments de mathématique.

The present text has defined a set to be finite if and only if there exists a bijection onto a natural number, and infinite if and only if there does not exist any such bijection.

2002, Yves Nievergelt, Foundations of Logic and Mathematics, page 214:

Note in particular that a function is a bijection if and only if it's both an injection and a surjection.

2007, C. J. Date, Logic and Databases: The Roots of Relational Theory, page 167:

The basic idea is that two sets A and B have the same cardinality if there is a bijection from A to B. Since the domain and range of the bijection is not relevant here, we often refer to a bijection from A to B as a bijection between the sets, or a one-to-one correspondence between the elements of the sets.

2013, William F. Basener, Topology and Its Applications, unnumbered page:

(set theory) A one-to-one correspondence, a function which is both a surjection and an injection.

Nicolas Bourbaki
bijection
Éléments de mathématique

биекция

function that is both a surjection and an injection


bijection

Je voudrais démontrer que cette fonction est une bijection réciproque.

I would like to show that this function is an inverse bijection.


surjection

/sɜː(ɹ)ˈd͡ʒɛk.ʃən/

LL-Q1860 (eng)-Flame, not lame-surjection.wav

From French surjection, introduced by Nicolas Bourbaki in their treatise Éléments de mathématique.
Ultimately borrowed from Latin superiectiō (“a throwing over or on; (fig.) an exaggeration, a hyperbole”).

bor

In some special cases, however, the number of surjections A#92;rightarrowB can be identified.

1992, Rowan Garnier, John Taylor, Discrete Mathematics for New Technology, Institute of Physics Publishing, page 220:

Let J#61;#92;cap#95;im#95;i be the (irredundant) primary decomposition of J. We associate to the pair (J,#92;omega) the element #92;textstyle#92;sum#95;i(m#95;i,#92;omega#95;i)#92;inG, where #92;omega#95;i is the equivalence class of surjections from L#47;m#95;iL#92;oplus(A#47;m#95;i)#123;n-1#125; to m#95;i#47;m#95;i² induced by #92;omega.

1999, M. Pavaman Murthy, “A survey of obstruction theory for projective modules of top rank”, in Tsit-Yuen Lam, Andy R. Magid, editors, Algebra, K-theory, Groups, and Education: On the Occasion of Hyman Bass's 65th Birthday, American Mathematical Society, page 168:

In Banach space theory, a mapping u#58;E#92;rightarrowF (between Banach spaces) is called a metric surjection if it is onto and if the associated mapping from E#47;#92;text#123;ker#125;(u) to F is an isometric isomorphism. Moreover, by the classical open mapping theorem, u is a surjection iff the associated mapping from E#47;#92;text#123;ker#125;(u) to F is an isomorphism.

2003, Gilles Pisier, Introduction to Operator Space Theory, Cambridge University Press, page 43:

(set theory) A function for which every element of the codomain is mapped to by some element of the domain; (formally) Any function f:X→Y for which for every y∈Y, there is at least one x∈X such that f(x)=y.

Nicolas Bourbaki
Surjective function
Éléments de mathématique


surjection

Borrowing from Latin superiectiōnem (“a throwing over or on; (figuratively) an exaggeration, a hyperbole”). Compare injection, bijection, with the same second element but different prefixes.


belong

en-verb

belong (third-person singular simple present belongs, present participle belonging, simple past and past participle belonged)

belongs (present singular third-person)
belonging (participle present)
belonged (participle past)
belonged (past)

⇒ Chinese Pidgin English: belong

→ Jersey Dutch: belânge

→ Tok Pisin: bilong

From Middle English belongen, bilongen, from Middle English be- + longen (“to be fitting, be suitable”), from Old English langian (“to pertain to, suit”), equivalent to be- + long (“to belong”). Compare Saterland Frisian beloangje (“to attain, reach, meet”), Dutch belangen (“to concern”), German belangen (“to sue, concern”).

cog

Suppose x belongs to #92;mathbb#123;R#125;…

(intransitive, set theory) (followed by to) To be an element of (a set). The symbol ∈ means belongs to.


creative

en-adj

creative (comparative more creative, superlative most creative)

more creative (comparative)
most creative (superlative)

Borrowed from Late Latin creativus, from Latin creō. Equivalent to create + -ive. Displaced native Old English orþanclīċ.

a creative set

(set theory) A type of set of natural numbers, related to mathematical logic.


normal

normal (comparative normaler or more normal, superlative normalest or most normal)

normaler (comparative)
more normal (comparative)
normalest (superlative)
most normal (superlative)

From Latin normālis (“made according to a carpenter's square; later: according to a rule”), from nōrma (“carpenter's square”), of uncertain origin; doublet of normale. The earliest meaning of the word in English was "perpendicular; forming a right angle" like something normālis (“made according to a carpenter's square”), but by Late Latin normālis had also come to mean "according to a rule", from which modern English senses of the word derive: in the 1800s, as people began to quantitatively study things like height, weight and blood pressure, the usual or most common values came to be called "normal", and by extension values regarded as healthy or desirable came to be called "normal" regardless of their usuality.

der

According to norms or rules or to a regular pattern.

(mathematics) Adhering to or being what is considered natural or regular in a particular field or context:

(set theory, of a function from the ordinals to the ordinals) Which is strictly monotonically increasing and continuous with respect to the order topology.


Use as a multiplication sign is usually attributed to English mathematician William Oughtred.

{1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)}

(set theory, category theory) Cartesian product, categorical product.


relation

en-noun

relation (countable and uncountable, plural relations)

From Middle English relacion, relacioun, from Anglo-Norman relacioun and Old French relacion (whence French relation), from Latin relātiō, noun of process form from perfect passive participle relātus (“related”), from verb referō (“I refer, I relate”), from prefix re- (“again”) + ferō (“I bear, I carry”). By surface analysis, relate + -ion. Doublet of relazione.

[…]Signs are, first of all, physical things: for example, chalk marks on a blackboard, pencil or ink marks on paper, sound waves produced in a human throat. According to Reichenbach, "What makes them signs is the intermediary position they occupy between an object and a sign user, i.e., a person." For a sign to be a sign, or to function as such, it is necessary that the person take account of the object it designates. Thus, anything in nature may or may not be a sign, depending on a person's attitude toward it. A physical thing is a sign when it appears as a substitute for, or representation of, the object for which it stands with respect to the sign user. The three-place relation between sign, object, and sign user is called the sign relation or relation of denotation.

1974, Thomas S. Szasz, M.D., chapter 7, in The Myth of Mental Illness, →ISBN, page 107:

Equality is a symmetric relation, while divisibility is not.

(set theory) Specifically, a set of ordered pairs; a binary relation.


filtration

filtration (countable and uncountable, plural filtrations)

(mathematics, set theory, probability theory) A totally ordered collection of subsets.


strict

en-adj

strict (comparative stricter or more strict, superlative strictest or most strict)

stricter (comparative)
more strict (comparative)
strictest (superlative)
most strict (superlative)

Borrowed from Latin strictus, past participle of stringere (“to draw tight, bind, contract”). Doublet of strait and stretto. See stringent, strain.

(set theory, order theory) Irreflexive; if the described object is defined to be reflexive, that condition is overridden and replaced with irreflexive.


collection

en-noun

collection (countable and uncountable, plural collections)

From Middle English colleccioun, collection, from Old French collection, from Latin collēctiō, collēctiōnem, from collēctus, from colligō (“collect together”), composed of con- + legō (“bring together, gather, collect”), ultimately from Proto-Indo-European *leǵ- (“to gather, collect”).

der

(set theory, topology, mathematical analysis) A set of sets; used because such a thing is in general too large to comply with the formal definition of a set.


correspondence

en-noun

correspondence (countable and uncountable, plural correspondences)

From Middle English correspondence, from Latin correspondentia. By surface analysis, correspond + -ence.

Congruity or similarity between different things, people, etc:


subset

proper subset

subset that is strictly less than the given other set

The set of integers is a subset of the set of real numbers.

The set #92;lbracea,b#92;rbrace is a both a subset and a proper subset of #92;lbracea,b,c#92;rbrace while the set #92;lbracea,b,c#92;rbrace is a subset of #92;lbracea,b,c#92;rbrace but not a proper subset of #92;lbracea,b,c#92;rbrace.

1963, David B. MacNeil, Modern Mathematics for the Practical Man, David Van Nostrand, Republished as 2013, David B. MacNeil, Fundamentals of Modern Mathematics: A Practical Review, Dover, page 3,
In the foregoing example, the set D of the first four letters of the alphabet, was a subset of the set A of all the letters of the alphabet, because A includes all the members of D.

Let A be a subset of the topological space X and take x#92;inX.

1997, Wolfgang Filter, K. Weber, Integration Theory, Chapman & Hall, page 5:

We say that a set S has a finite partition into subsets S#95;1,#92;dots,S#95;n, if S#61;S#95;i#92;cup#92;dots#92;cupS#95;n, where the subsets are pairwise disjoint, that is, S#95;i#92;capS#95;j#61;#92;empty, if i#92;nej. (We do not require that the subsets be nonempty.)

2007, Judith D. Sally, Paul J. Sally, Jr., Roots to Research: A Vertical Development of Mathematical Problems, American Mathematical Society, page 280:

(set theory, of a set S) A set A such that every element of A is also an element of S.

подмножество

set whose elements are within another given set


forcing

forcing (countable and uncountable, plural forcings)

(set theory) A technique used to prove the consistency of certain axioms in set theory. See forcing (mathematics).

כפייה

set theory: a technique used to prove the consistency of certain axioms


uguale

uguali (plural)
eguale (alternative uncommon)
equale (alternative archaic)

From earlier equale, from Latin aequālis, derived from aequus (“equal, even”).

(set theory) containing the same elements


successor

successors (plural)
successour (alternative obsolete)

predecessor

person or thing that immediately follows another

From Anglo-Norman successour, from Latin successor.

A limit ordinal is not the successor of any ordinal.

(arithmetic, set theory) The integer, ordinal number or cardinal number immediately following another.

наследник

integer or cardinal immediately following another


aleph-null

From its mathematical symbol ℵ₀, from א, aleph, the first letter of the Hebrew alphabet.

(set theory) The first of the transfinite cardinal numbers; corresponds to the number of natural numbers.


computable

(mathematics, set theory) Of a countably infinite set, having a computable indicator function.


saturated

en-adj

saturated (comparative more saturated, superlative most saturated)

more saturated (comparative)
most saturated (superlative)

LL-Q1860 (eng)-Flame, not lame-saturated.wav

(mathematics, set theory, of a set S, with respect to an equivalence relation ≡) Equal to a union of equivalence classes of ≡; such that if x∈S and x≡y then y∈S.

(mathematics, especially topology, with respect to a function f) Saturated with respect to the equivalence relation x≡y⟺f(x)=f(y) (the kernel, sense 10, of f).


transfinite number

en-noun

transfinite number (plural transfinite numbers)

It will be recalled that Cantor called the first transfinite number ℵ₀. He called the second transfinite number—the one describing the set of all real numbers— C. It has not been proved whether C is the next transfinite number after ℵ₀ or whether another number exists between them.

1961, Jane Muir, Of Men and Numbers: The Story of the Great Mathematicians, Courier Dover Publications, →ISBN, page 228:

Let R be a bicompact of dimensionality #92;operatorname#123;ind#125;(R)#92;le#92;alpha. If #92;alpha is an isolated transfinite number, than ^([sic]) at any point x#92;inR there exist arbitrarily small neighborhoods Vx with boundaries of dimensionality #92;operatorname#123;ind#125;#92;overline#123;Vx#125;#92;le#92;alpha-1.

1968, B. T. Levšenko, “Spaces of transfinite dimensionality”, in Fourteen Papers on Algebra, Topology, Algebraic and Differential Geometry, American Mathematical Soc., →ISBN, page 141:

After all, it was the ordinals that made precise definition of the transfinite cardinals possible. And until Cantor had introduced the order types of transfinite number classes, he could not define precisely any transfinite cardinal beyond the first power.

1990, Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton University Press, →ISBN, page 180:

For example, does there exist a transfinite number that is strictly bigger than ℵ₀ and strictly smaller than ℵ₁? In this case an instance of this in between number is too big to be put into one-to-one correspondence with the set of natural numbers, and too small to be put into one-to-one correspondence with the set of real numbers.

2009, John Tabak, Numbers: Computers, Philosophers, and the Search for Meaning, Infobase Publishing, →ISBN, page 153:

Having demonstrated the existence of a one-to-one correspondence, we can conclude that the class of the squares of all the natural numbers has the same transfinite number as the class of all the natural numbers! This result is not what might have been anticipated, seeing that the second class is a proper subset of the first.

2012, Benjamin Wardhaugh, A Wealth of Numbers: An Anthology of 500 Years of Popular Mathematics Writing, Princeton University Press, →ISBN, page 136:

(set theory) Any cardinal or ordinal number which is larger than any finite, i.e. natural number; often represented by the Hebrew letter aleph (ℵ) with a subscript 0, 1, etc.


aleph-one

continuum hypothesis
continuum hypothesis

(set theory) The second of the transfinite cardinal numbers; according to the continuum hypothesis, it corresponds to the number of real numbers.


partition

partition (countable and uncountable, plural partitions)

/pɑɹˈtɪ.ʃən/

/pɑɹˈtɪ.ʃɪn/

Recorded c.1430, "division into shares, distinction," from Middle English particioun, from Old French particion (modern partition), from Latin partitio, partitionem (“division, portion”), from partitus, the past participle of partire (“to split (up), part(ition)”).

der

(set theory) A collection of non-empty, disjoint subsets of a set whose union is the set itself (i.e. all elements of the set are contained in exactly one of the subsets).


ℵ

Introduced by Georg Cantor, based on the Hebrew letter א, initial letter of אינסוף (infinity).

(set theory) Symbol for the transfinite enumeration of the infinite cardinal numbers in order of magnitude, a function mapping ordinal numbers to infinite cardinals.


множина

uk-noun

множина́ • (množyná) f inan (genitive множини́, nominative plural множи́ни, genitive plural множи́н, relational adjective множи́нний)

множина́ (canonical)
množyná (romanization)
множини́ (genitive)
множи́ни (nominative plural)
множи́н (genitive plural)
множи́нний (adjective relational)
множина́ (nominative singular)
множи́ни (nominative plural)
множини́ (genitive singular)
множи́н (genitive plural)
множині́ (dative singular)
множи́нам (dative plural)
множину́ (accusative singular)
множи́ни (accusative plural)
множино́ю (instrumental singular)
множи́нами (instrumental plural)
множині́ (locative singular)
множи́нах (locative plural)
множи́но (singular vocative)
множи́ни (plural vocative)

Uk-множина.ogg

uk:Множина
uk:Множина (мовознавство)


množina

množina (nominative singular)
množiny (nominative plural)
množiny (genitive singular)
množin (genitive plural)
množině (dative singular)
množinám (dative plural)
množinu (accusative singular)
množiny (accusative plural)
množino (singular vocative)
množiny (plural vocative)
množině (locative singular)
množinách (locative plural)
množinou (instrumental singular)
množinami (instrumental plural)


množina

sk-noun

množina f (relational adjective množinový)

množinový (adjective relational)
singular
plural
nominative
množina
množiny
genitive
množín
dative
množine
množinám
accusative
množinu
locative
množinách
instrumental
množinou
množinami


L

From the Etruscan letter 𐌋 (l, “el”), from the Ancient Greek letter Λ (L, “lambda”), derived from the Phoenician letter 𐤋 (l, “lamed”), from the Egyptian hieroglyph 𓌅.


S

From the first letter of set and of stage, two fundamental concepts of the theory.

S has no axiom of extensionality and identity is absent from the other S axioms.

(set theory) An axiomatic set theory, developed by logician George Boolos, in which several of the axioms of ZF are derivable as theorems.


V

From the Etruscan letter 𐌖 (u, “u”), from the Ancient Greek letter Υ (U, “ypsilon”), derived from the Phoenician letter 𐤅 (w, “waw”), from the Egyptian hieroglyph 𓏲. Doublet of Y, U, and F.

v (alternative numeral)
V. (alternative numeral ordinal)


tuple

LL-Q1860 (eng)-Flame, not lame-tuple.wav

From the ending of the words quintuple, sextuple; from Latin -plus.

(set theory) A finite sequence of terms.


singleton

singleton (nominative singular)
singletony (nominative plural)
singletonu (genitive singular)
singletonów (genitive plural)
singletonowi (dative singular)
singletonom (dative plural)
singleton (accusative singular)
singletony (accusative plural)
singletonem (instrumental singular)
singletonami (instrumental plural)
singletonie (locative singular)
singletonach (locative plural)
singletonie (singular vocative)
singletony (plural vocative)

(set theory) singleton, unit set, one-point set (set with exactly one element)


ZF

Gödel's model is an example of a simple type of inner model that might be called a definable transitive inner model³), where the universe is replaced by a transitive subuniverse defined within ZF, the membership relation is just the original one restricted to the subuniverse, and where the axioms of ZF relativized to the subuniverse, are provable within ZF.

1967, Carol Karp, “A Proof of the Relative Consistency of the Continuum Hypothesis”, in John N. Crossley, editor, Sets, Models, and Recursion Theory, North-Holland, page 7:

1971, Ulrich Felgner, Models of ZF-Set Theory, Springer, Lecture Notes in Mathematics 223, page 21,
1. Corollary: ZF is not finitely axiomatizable.
2. Corollary: ZF is reflexive (i.e. the consistency of every finite subtheory of ZF can be proved within ZF).

1991 [Kluwer Academic], Fred Landman, Structures for Semantics, 1991, Springer, Softcover, page 56,
However, the problem with it, and the reason why it is not part of ZF strictly (apart from the fact that it implies the axiom of choice) is that it is rather arbitrary.

Zermelo-Fraenkel
w:Zermelo-Fraenkel set theory

(set theory) Initialism of Zermelo-Fraenkel (set theory): a particular axiomatic formulation of set theory without the axiom of choice.


superset

LL-Q1860 (eng)-Flame, not lame-superset.wav

The set of human beings is a superset of the set of human children.

The set of characters "LBPG" is a superset of the set of characters "PG".

(set theory) (symbol: ⊇) With respect to another set, a set such that each of the elements of the other set is also an element of the set.

nadmnožina

set containing all elements of another set


perhe

perhe (nominative singular)
perheet (nominative plural)
perhe (accusative nominative singular)
perheet (accusative nominative plural)
perheen (accusative genitive singular)
perheet (accusative genitive plural)
perheen (genitive singular)
perheiden (genitive plural)
perheitten (genitive plural)
perhettä (partitive singular)
perheitä (partitive plural)
perheessä (inessive singular)
perheissä (inessive plural)
perheestä (elative singular)
perheistä (elative plural)
perheeseen (illative singular)
perheisiin (illative plural)
perheihin (illative plural)
perheellä (adessive singular)
perheillä (adessive plural)
perheeltä (ablative singular)
perheiltä (ablative plural)
perheelle (allative singular)
perheille (allative plural)
perheenä (essive singular)
perheinä (essive plural)
perheeksi (singular translative)
perheiksi (plural translative)
perheettä (abessive singular)
perheittä (abessive plural)
perhein (instructive plural)
See the possessive forms below. (comitative)
perheeni (nominative singular)
perheeni (nominative plural)
perheeni (accusative nominative singular)
perheeni (accusative nominative plural)
perheeni (accusative genitive singular)
perheeni (accusative genitive plural)
perheeni (genitive singular)
perheideni (genitive plural)
perheitteni (genitive plural)
perhettäni (partitive singular)
perheitäni (partitive plural)
perheessäni (inessive singular)
perheissäni (inessive plural)
perheestäni (elative singular)
perheistäni (elative plural)
perheeseeni (illative singular)
perheisiini (illative plural)
perheihini (illative plural)
perheelläni (adessive singular)
perheilläni (adessive plural)
perheeltäni (ablative singular)
perheiltäni (ablative plural)
perheelleni (allative singular)
perheilleni (allative plural)
perheenäni (essive singular)
perheinäni (essive plural)
perheekseni (singular translative)
perheikseni (plural translative)
perheettäni (abessive singular)
perheittäni (abessive plural)
perheineni (comitative plural)
perheesi (nominative singular)
perheesi (nominative plural)
perheesi (accusative nominative singular)
perheesi (accusative nominative plural)
perheesi (accusative genitive singular)
perheesi (accusative genitive plural)
perheesi (genitive singular)
perheidesi (genitive plural)
perheittesi (genitive plural)
perhettäsi (partitive singular)
perheitäsi (partitive plural)
perheessäsi (inessive singular)
perheissäsi (inessive plural)
perheestäsi (elative singular)
perheistäsi (elative plural)
perheeseesi (illative singular)
perheisiisi (illative plural)
perheihisi (illative plural)
perheelläsi (adessive singular)
perheilläsi (adessive plural)
perheeltäsi (ablative singular)
perheiltäsi (ablative plural)
perheellesi (allative singular)
perheillesi (allative plural)
perheenäsi (essive singular)
perheinäsi (essive plural)
perheeksesi (singular translative)
perheiksesi (plural translative)
perheettäsi (abessive singular)
perheittäsi (abessive plural)
perheinesi (comitative plural)
perheemme (nominative singular)
perheemme (nominative plural)
perheemme (accusative nominative singular)
perheemme (accusative nominative plural)
perheemme (accusative genitive singular)
perheemme (accusative genitive plural)
perheemme (genitive singular)
perheidemme (genitive plural)
perheittemme (genitive plural)
perhettämme (partitive singular)
perheitämme (partitive plural)
perheessämme (inessive singular)
perheissämme (inessive plural)
perheestämme (elative singular)
perheistämme (elative plural)
perheeseemme (illative singular)
perheisiimme (illative plural)
perheihimme (illative plural)
perheellämme (adessive singular)
perheillämme (adessive plural)
perheeltämme (ablative singular)
perheiltämme (ablative plural)
perheellemme (allative singular)
perheillemme (allative plural)
perheenämme (essive singular)
perheinämme (essive plural)
perheeksemme (singular translative)
perheiksemme (plural translative)
perheettämme (abessive singular)
perheittämme (abessive plural)
perheinemme (comitative plural)
perheenne (nominative singular)
perheenne (nominative plural)
perheenne (accusative nominative singular)
perheenne (accusative nominative plural)
perheenne (accusative genitive singular)
perheenne (accusative genitive plural)
perheenne (genitive singular)
perheidenne (genitive plural)
perheittenne (genitive plural)
perhettänne (partitive singular)
perheitänne (partitive plural)
perheessänne (inessive singular)
perheissänne (inessive plural)
perheestänne (elative singular)
perheistänne (elative plural)
perheeseenne (illative singular)
perheisiinne (illative plural)
perheihinne (illative plural)
perheellänne (adessive singular)
perheillänne (adessive plural)
perheeltänne (ablative singular)
perheiltänne (ablative plural)
perheellenne (allative singular)
perheillenne (allative plural)
perheenänne (essive singular)
perheinänne (essive plural)
perheeksenne (singular translative)
perheiksenne (plural translative)
perheettänne (abessive singular)
perheittänne (abessive plural)
perheinenne (comitative plural)
perheensä (nominative singular)
perheensä (nominative plural)
perheensä (accusative nominative singular)
perheensä (accusative nominative plural)
perheensä (accusative genitive singular)
perheensä (accusative genitive plural)
perheensä (genitive singular)
perheidensä (genitive plural)
perheittensä (genitive plural)
perhettään (partitive singular)
perhettänsä (partitive singular)
perheitään (partitive plural)
perheitänsä (partitive plural)
perheessään (inessive singular)
perheessänsä (inessive singular)
perheissään (inessive plural)
perheissänsä (inessive plural)
perheestään (elative singular)
perheestänsä (elative singular)
perheistään (elative plural)
perheistänsä (elative plural)
perheeseensä (illative singular)
perheisiinsä (illative plural)
perheihinsä (illative plural)
perheellään (adessive singular)
perheellänsä (adessive singular)
perheillään (adessive plural)
perheillänsä (adessive plural)
perheeltään (ablative singular)
perheeltänsä (ablative singular)
perheiltään (ablative plural)
perheiltänsä (ablative plural)
perheelleen (allative singular)
perheellensä (allative singular)
perheilleen (allative plural)
perheillensä (allative plural)
perheenään (essive singular)
perheenänsä (essive singular)
perheinään (essive plural)
perheinänsä (essive plural)
perheekseen (singular translative)
perheeksensä (singular translative)
perheikseen (plural translative)
perheiksensä (plural translative)
perheettään (abessive singular)
perheettänsä (abessive singular)
perheittään (abessive plural)
perheittänsä (abessive plural)
perheineen (comitative plural)
perheinensä (comitative plural)

Through metathesis from Proto-Finnic *pereh, probably derived from *perä (“rear”), referring to the background of a person (compare alkuperä (“origin”), peräisin (“originating from”), perehtyä). First written appearance is from 1548 in the Finnish version of Sweden's maanlaki (“law of the country”) translated by a cleric known as Herra Martti.
Cognate with Estonian pere, Karelian pereh, Livonian perē, Veps pereh and Võro pereh.


empty set

Start with the empty set. We define it as “the set of all objects not equal to themselves,” since there are no such objects. All empty sets have the same members—no members at all! Therefore, as sets they're identical, by definition of identity of sets. In other words, there's only one empty set. This unique empty set is our building block. Next comes the set whose only member is—the empty set. This set is not empty.

1997, Reuben Hersh, What Is Mathematics, Really?, Oxford University Press, page 257:

The existence of the empty set is entailed, in the presence of the axiom scheme of separation, by the existence of any set whatever.

2004, Michael Potter, Set Theory and its Philosophy, Oxford University Press, page 58:

A topological space may be very sparsely endowed with open sets. As we know, some spaces have only two, the empty set and the full space.

2007, A. K. Sharma, Measure Theory, Discovery Publishing House, page 1:

Their existence can be generated on the basis of the empty set, since one is simply the set of the empty set, two is simply that set plus the empty set and so forth.

2008, Sam Gillespie, The Mathematics of Novelty: Badiou's Minimalist Metaphysics, re.press, page 57:

(set theory) The unique set that contains no elements, denoted ∅ or {}.


cardinality

en-noun

cardinality (countable and uncountable, plural cardinalities)

The empty set has a cardinality of zero.

The cardinality of a set A is the least ordinal α such that there exists a bijection between A and α. We sometimes use the notation #92;alpha#61;#124;A#124; to indicate this.

2005, Johan de Jong, “Set Theory”, in The Stacks Project, retrieved 2018-02-26:

For fuzzy sets, the concept of set size or cardinality is both richer and more problematic than it is for crisp sets. It is richer because, as we shall see, we may use more than one kind of cardinality.

2006, Michael Smithson, Jay Verkuilen, Fuzzy Set Theory: Applications in the Social Sciences, SAGE Publications, page 37:

Clearly, in this example, the sensitivity to the cardinalities takes the weaker form F#91;M(A)#93;#61;#92;overset#123;#61;#125;#123;A#125; of a single-valued function from the measure to the cardinality rather than the stronger form M(A)#61;f(#92;overset#123;#61;#125;#123;A#125;) of a function from the cardinality to the measure.

2012, Adolf Grünbaum, Robert S. Cohen, Marx W. Wartofsky, Philosophical Problems of Space and Time, 2nd edition, Springer, page 487:

(set theory, of a set) The number of elements a given set contains.


power set

power sets (plural)
power-set (alternative)
powerset (alternative)

The power set of #92;#123;1,2#92;#125; is #92;left#92;#123;#92;empty,#92;#123;1#92;#125;,#92;#123;2#92;#125;,#92;#123;1,2#92;#125;#92;right#92;#125;.

Moreover, for notational convenience, we write the cardinality of a denumerable set as #92;aleph#95;0. Cardinality of the power set of a denumerable set is written as #92;aleph#95;1. We may thus extend this notation further by taking cardinality of the power set of the power set of a denumerable set as #92;aleph#95;2, etc. but we do not have the need for it right now.

2009, Arindama Singh, Elements of Computation Theory, Springer, page 16:

Theorem 4.1. A complete Boolean algebra B has a set of (complete and atomic) ca-free generators iff B is isomorphic to the power set of a power set.

2013, A. Carsetti, Epistemic Complexity and Knowledge Construction, Springer, page 94:

Exponentiation is essentially a move to the power set—the set of all subsets of a given set. This is one of the reasons why Bertrand Russell's paradox is indeed a paradox: We cannot find a universal set because no set can contain its own power set!

2015, Amir D. Aczel, Finding Zero: A Mathematician's Odyssey to Uncover the Origins of Numbers, Palgrave MacMillan, page 147:

(set theory, of a set S) The set whose elements comprise all the subsets of S (including the empty set and S itself).


null set

(set theory, sometimes proscribed) The empty set.


negligible set

(mathematics, set theory, measure theory) A set that is small enough that it can be ignored for some purpose.


universal set

(set theory) A set large enough to contain all sets under consideration in the current context.


proper subset

{2, 4, 5} is a proper subset of {1, 2, 3, 4, 5}, but {1, 2, 3, 4, 5} is not.

(set theory) a set that is a subset of the set, but not equal to it.

真子集

a set that is a subset of but not equal to another set


Venn diagram

Named after British mathematician and philosopher John Venn (1834–1923).

named-after

(set theory) A diagram representing some sets by contours of closed shapes, such as circles or ellipses (and sometimes also the universal set as a rectangle enclosing all of these shapes), and indicating the relationships between the sets: by overlapping the shapes to show that the corresponding sets have a non-empty intersection, and by possibly (but not necessarily) enclosing all of the sets (which are proper subsets of the universal set) within a universal set (represented typically by a rectangle); such that the total number of simply connected regions is 2ⁿ, where n is the number of depicted sets which are proper subsets of the universal set.


non-empty

(set theory) Of a set, containing at least one element; not the empty set.


disjoint

en-adj

disjoint (comparative more disjoint, superlative most disjoint)

more disjoint (comparative)
most disjoint (superlative)

From Middle English disjoynen, from Old French desjoindre (“disjoin”), from Latin disiungō, from dis- + iungō (“join”).

(set theory, not comparable) Of two or more sets, having no members in common; having an intersection equal to the empty set.


ordered pair

If an ordered pair were defined (in terms of sets) as (x,y):=a,a,b then the "first element" of an ordered pair S could be defined as CAR(S) where CAR(S) = x if and only if (∀y∈S.,x∈y). Likewise, the "second element" of S could be defined as CDR(S) where CDR(S) = x if and only if (∃y∈S.,(∃z∈y.,x∈z)). If the two elements happened to be equal, then the ordered pair would still have cardinality two as would be naturally expected.

(set theory) An object containing exactly two elements in a fixed order, so that, when the elements are different, exchanging them gives a different object. Notation: (a, b) or ⟨a,b⟩.


equivalence relation

en-noun

equivalence relation (plural equivalence relations)

(set theory) A binary relation that is reflexive, symmetric and transitive.


binary relation

binary relation (plural binary relations)

A partially ordered set #92;langleA,#92;varrho#92;rangle consists of a nonvoid set A and a binary relation #92;varrho on A, such that #92;varrho satisfies properties (P1)-(P3).

1978, George Grätzer, General Lattice Theory, Academic Press, page 1:

1.30. Corollary. If P is a binary relation which is asymmetric and negatively transitive, then P is also transitive.
It should be noted that a binary relation may be irreflexive and negatively transitive without being transitive; as an example, consider the standard inequality relation (≠).

1999, James C. Moore, Mathematical Methods for Economic Theory 1, Springer, page 24:

Definition If E is a non-empty set then by an order on E we mean a binary relation on E that is reflexive, anti-symmetric, and transitive.

2005, T. S. Blyth, Lattices and Ordered Algebraic Structures, Springer, page 1:

(set theory, order theory, "on" a set A) A subset of the Cartesian product A×A (the set of ordered pairs (a, b) of elements of A).

(set theory, order theory, "on" or "between" sets A and B) A subset of the Cartesian product A×B.


setoid

(set theory) A set together with an equivalence relation.


equivalence class

equivalence class (plural equivalence classes)

(set theory) Any one of the subsets into which an equivalence relation partitions a set, each of these subsets containing all the elements of the set that are equivalent under the equivalence relation.

třída ekvivalence

Any one of the subsets into which an equivalence relation partition is set


productive

en-adj

productive (comparative more productive, superlative most productive)

more productive (comparative)
most productive (superlative)

Borrowed from Late Latin productivus, equivalent to product + -ive.

a productive set

(set theory) A type of set of natural numbers, related to mathematical logic; a set S is productive if there exists a total recursive function f such that ∀x∈ℕ,~~W_x⊆S⇒f(x)∈A⧵W_x, where W_x is a recursive function whose Gödel number is x.


íbúi

is-noun

íbúi m (genitive singular íbúa, nominative plural íbúar)

íbúa (genitive singular)
íbúar (nominative plural)
íbúi (indefinite nominative singular)
íbúinn (definite nominative singular)
íbúar (indefinite nominative plural)
íbúarnir (definite nominative plural)
íbúa (accusative indefinite singular)
íbúann (accusative definite singular)
íbúa (accusative indefinite plural)
íbúana (accusative definite plural)
íbúa (dative indefinite singular)
íbúanum (dative definite singular)
íbúum (dative indefinite plural)
íbúunum (dative definite plural)
íbúa (genitive indefinite singular)
íbúans (definite genitive singular)
íbúa (genitive indefinite plural)
íbúanna (definite genitive plural)

(set theory) an element, a member; (one of the objects in a set)


saturation

en-noun

saturation (usually uncountable, plural saturations)

LL-Q1860 (eng)-Flame, not lame-saturation.wav

Etymology tree
Late Latin saturatiobor.
English saturation
Borrowed from Late Latin saturatio, saturationem.

etymon

(mathematics, set theory, topology, of a set S, with respect to an equivalence relation or function, countable) The smallest set containing S which is saturated with respect to the equivalence relation or function.


meager

en-adj

meager (comparative meagerer, superlative meagerest)

meagerer (comparative)
meagerest (superlative)
meagre (alternative Commonwealth)

Jamaican Creole: mawga

From Middle English megre, from Anglo-Norman megre, Old French maigre, from Latin macer, from Proto-Indo-European *mh₂ḱrós. Akin, through the Indo-European root, to Old English mæġer (“meager, lean”), West Frisian meager (“meager”), Dutch mager (“meager”), German mager, Icelandic magr whence the Icelandic magur,
Norwegian Bokmål mager and Danish mager. Doublet of maigre.

(set theory) Of a set: such that, considered as a subset of a (usually larger) topological space, it is in a precise sense small or negligible.


元素

/ɥɛn³⁵ su⁵¹/

/ᵑɡun²² ɬu³³/

/ŋœŋ¹³⁻⁵⁵ ɬɔu⁴²/

/ŋyøŋ²⁴⁻⁵⁵ ɬɔu⁴²/

/ɡuan²⁴⁻²² sɔ²¹/

/ɡuan²⁴⁻²² sɔ⁴¹/

/ɡuan¹³⁻²² sɔ²¹/

/ɡuan²⁴⁻¹¹ sɔ¹¹/

/ɡuan²³⁻³³ sɔ²¹/

/ŋueŋ⁵⁵⁻¹¹ su²¹³/

/ŋuaŋ⁵⁵⁻¹¹ su²¹³/

/ɥɛn³⁵ su⁵¹/

/ᵑɡun²² ɬu³³/

/ŋœŋ¹³⁻⁵⁵ ɬɔu⁴²/

/ŋyøŋ²⁴⁻⁵⁵ ɬɔu⁴²/

/ɡuan²⁴⁻²² sɔ²¹/

/ɡuan²⁴⁻²² sɔ⁴¹/

/ɡuan¹³⁻²² sɔ²¹/

/ɡuan²⁴⁻¹¹ sɔ¹¹/

/ɡuan²³⁻³³ sɔ²¹/

/ŋueŋ⁵⁵⁻¹¹ su²¹³/

/ŋuaŋ⁵⁵⁻¹¹ su²¹³/

(set theory) element (one of the objects in a set) (Classifier: 個／个 m)


Θ

(set theory) Θ, the ordinal number with the same cardinality as the smallest set larger than the real numbers; the smallest nonzero ordinal number such that there exists no surjection onto it from the real numbers.


Δ

symmetric difference
symmetric difference

(mathematics, set theory) Used to represent the symmetric difference (also known as the disjunctive union) of two sets


ω

(mathematics, set theory) The first (countably) infinite ordinal number, its corresponding cardinal number ℵ₀ or the set of natural numbers (the latter of which are often defined to equal the former).


directed

directed (comparative more directed, superlative most directed)

more directed (comparative)
most directed (superlative)

(set theory) Having the properties of a directed set.


verzameling

nl-noun

verzameling f (plural verzamelingen, diminutive verzamelingetje n)

verzamelingen (plural)
verzamelingetje (diminutive neuter)

Afrikaans: versameling

desc

Negerhollands: versammeling, versamling, versammling

→ Indonesian: himpunan (semantic loan)

From Middle Dutch versamelinge. Equivalent to verzamelen + -ing.

Verzamelingen zijn een belangrijk concept in de verzamelingenleer binnen de wiskunde.

Sets are an important concept in set theory within mathematics.


meagre

en-adj

meagre (comparative meagrer, superlative meagrest)

meagrer (comparative)
meagrest (superlative)
meager (alternative)

Jamaican Creole: mawga

From Middle English megre, borrowed from Anglo-Norman megre, Old French maigre, from Latin macer, macrum, from Proto-Indo-European *mh₂ḱrós. Cognate with Old English mæġer (“meagre, lean”), Dutch mager (“lean”), German mager (“lean”), Icelandic magur (“lean”).


injection

en-noun

injection (countable and uncountable, plural injections)

/ɪnˈd͡ʒɛk.ʃən/

Nicolas Bourbaki
Éléments de mathématique

Borrowed from Middle French injection, from Latin iniectio. The mathematical sense is from French injection, introduced by Nicolas Bourbaki in their treatise Éléments de mathématique.

(set theory) A function that maps distinct x in the domain to distinct y in the codomain; formally, a f: X → Y such that f(a) = f(b) implies a = b for any a, b in the domain.


injection

→ Romanian: injecție

→ Turkish: enjeksiyon

(mathematics, logic, set theory) injection


bounded

en-adj

bounded (comparative more bounded, superlative most bounded)

more bounded (comparative)
most bounded (superlative)

(set theory, order theory, of a poset X with partial order ≤) That contains a least element, a, and a greatest element, b, such that for all x ∈ X, a ≤ x ≤ b.

barita

that contains a least and a greatest elements


The set-theoretic meaning of this symbol was introduced by the Bourbaki group in their treatise Éléments de mathématique. André Weil has said in his autobiography that he based it on the Norwegian letter Ø.

(set theory) A set with no elements: the empty set.


kunta

kunta (nominative singular)
kunnat (nominative plural)
kunta (accusative nominative singular)
kunnat (accusative nominative plural)
kunnan (accusative genitive singular)
kunnat (accusative genitive plural)
kunnan (genitive singular)
kuntien (genitive plural)
kuntain (genitive plural rare)
kuntaa (partitive singular)
kuntia (partitive plural)
kunnassa (inessive singular)
kunnissa (inessive plural)
kunnasta (elative singular)
kunnista (elative plural)
kuntaan (illative singular)
kuntiin (illative plural)
kunnalla (adessive singular)
kunnilla (adessive plural)
kunnalta (ablative singular)
kunnilta (ablative plural)
kunnalle (allative singular)
kunnille (allative plural)
kuntana (essive singular)
kuntina (essive plural)
kunnaksi (singular translative)
kunniksi (plural translative)
kunnatta (abessive singular)
kunnitta (abessive plural)
kunnin (instructive plural)
See the possessive forms below. (comitative)
kuntani (nominative singular)
kuntani (nominative plural)
kuntani (accusative nominative singular)
kuntani (accusative nominative plural)
kuntani (accusative genitive singular)
kuntani (accusative genitive plural)
kuntani (genitive singular)
kuntieni (genitive plural)
kuntaini (genitive plural rare)
kuntaani (partitive singular)
kuntiani (partitive plural)
kunnassani (inessive singular)
kunnissani (inessive plural)
kunnastani (elative singular)
kunnistani (elative plural)
kuntaani (illative singular)
kuntiini (illative plural)
kunnallani (adessive singular)
kunnillani (adessive plural)
kunnaltani (ablative singular)
kunniltani (ablative plural)
kunnalleni (allative singular)
kunnilleni (allative plural)
kuntanani (essive singular)
kuntinani (essive plural)
kunnakseni (singular translative)
kunnikseni (plural translative)
kunnattani (abessive singular)
kunnittani (abessive plural)
kuntineni (comitative plural)
kuntasi (nominative singular)
kuntasi (nominative plural)
kuntasi (accusative nominative singular)
kuntasi (accusative nominative plural)
kuntasi (accusative genitive singular)
kuntasi (accusative genitive plural)
kuntasi (genitive singular)
kuntiesi (genitive plural)
kuntaisi (genitive plural rare)
kuntaasi (partitive singular)
kuntiasi (partitive plural)
kunnassasi (inessive singular)
kunnissasi (inessive plural)
kunnastasi (elative singular)
kunnistasi (elative plural)
kuntaasi (illative singular)
kuntiisi (illative plural)
kunnallasi (adessive singular)
kunnillasi (adessive plural)
kunnaltasi (ablative singular)
kunniltasi (ablative plural)
kunnallesi (allative singular)
kunnillesi (allative plural)
kuntanasi (essive singular)
kuntinasi (essive plural)
kunnaksesi (singular translative)
kunniksesi (plural translative)
kunnattasi (abessive singular)
kunnittasi (abessive plural)
kuntinesi (comitative plural)
kuntamme (nominative singular)
kuntamme (nominative plural)
kuntamme (accusative nominative singular)
kuntamme (accusative nominative plural)
kuntamme (accusative genitive singular)
kuntamme (accusative genitive plural)
kuntamme (genitive singular)
kuntiemme (genitive plural)
kuntaimme (genitive plural rare)
kuntaamme (partitive singular)
kuntiamme (partitive plural)
kunnassamme (inessive singular)
kunnissamme (inessive plural)
kunnastamme (elative singular)
kunnistamme (elative plural)
kuntaamme (illative singular)
kuntiimme (illative plural)
kunnallamme (adessive singular)
kunnillamme (adessive plural)
kunnaltamme (ablative singular)
kunniltamme (ablative plural)
kunnallemme (allative singular)
kunnillemme (allative plural)
kuntanamme (essive singular)
kuntinamme (essive plural)
kunnaksemme (singular translative)
kunniksemme (plural translative)
kunnattamme (abessive singular)
kunnittamme (abessive plural)
kuntinemme (comitative plural)
kuntanne (nominative singular)
kuntanne (nominative plural)
kuntanne (accusative nominative singular)
kuntanne (accusative nominative plural)
kuntanne (accusative genitive singular)
kuntanne (accusative genitive plural)
kuntanne (genitive singular)
kuntienne (genitive plural)
kuntainne (genitive plural rare)
kuntaanne (partitive singular)
kuntianne (partitive plural)
kunnassanne (inessive singular)
kunnissanne (inessive plural)
kunnastanne (elative singular)
kunnistanne (elative plural)
kuntaanne (illative singular)
kuntiinne (illative plural)
kunnallanne (adessive singular)
kunnillanne (adessive plural)
kunnaltanne (ablative singular)
kunniltanne (ablative plural)
kunnallenne (allative singular)
kunnillenne (allative plural)
kuntananne (essive singular)
kuntinanne (essive plural)
kunnaksenne (singular translative)
kunniksenne (plural translative)
kunnattanne (abessive singular)
kunnittanne (abessive plural)
kuntinenne (comitative plural)
kuntansa (nominative singular)
kuntansa (nominative plural)
kuntansa (accusative nominative singular)
kuntansa (accusative nominative plural)
kuntansa (accusative genitive singular)
kuntansa (accusative genitive plural)
kuntansa (genitive singular)
kuntiensa (genitive plural)
kuntainsa (genitive plural rare)
kuntaansa (partitive singular)
kuntiaan (partitive plural)
kuntiansa (partitive plural)
kunnassaan (inessive singular)
kunnassansa (inessive singular)
kunnissaan (inessive plural)
kunnissansa (inessive plural)
kunnastaan (elative singular)
kunnastansa (elative singular)
kunnistaan (elative plural)
kunnistansa (elative plural)
kuntaansa (illative singular)
kuntiinsa (illative plural)
kunnallaan (adessive singular)
kunnallansa (adessive singular)
kunnillaan (adessive plural)
kunnillansa (adessive plural)
kunnaltaan (ablative singular)
kunnaltansa (ablative singular)
kunniltaan (ablative plural)
kunniltansa (ablative plural)
kunnalleen (allative singular)
kunnallensa (allative singular)
kunnilleen (allative plural)
kunnillensa (allative plural)
kuntanaan (essive singular)
kuntanansa (essive singular)
kuntinaan (essive plural)
kuntinansa (essive plural)
kunnakseen (singular translative)
kunnaksensa (singular translative)
kunnikseen (plural translative)
kunniksensa (plural translative)
kunnattaan (abessive singular)
kunnattansa (abessive singular)
kunnittaan (abessive plural)
kunnittansa (abessive plural)
kuntineen (comitative plural)
kuntinensa (comitative plural)

From Proto-Finnic *kunta, from Proto-Finno-Ugric *kunta. Cognates include Estonian -kond, Livonian -gõnd (kōzgõnd), Votic kuntõ, Erzya кондямо (konďamo), Moksha кондя (kondä), Northern Mansi [script needed] (χōnt) and Hungarian had.


Cartesian product

en-noun

Cartesian product (plural Cartesian products)

Cartesian products (plural)
cartesian product (alternative)

From Cartesian + product, after French philosopher, mathematician, and scientist René Descartes (1596–1650), whose formulation of analytic geometry gave rise to the concept.

(set theory, of two sets X and Y) The set of all possible ordered pairs of elements, the being first from X, the second from Y, written X×Y. Formally, the set (x,y);|;x∈X;and;y∈Y.


skup

skȕp (canonical masculine)
ску̏п (Cyrillic)
skup (nominative singular)
skupovi (nominative plural)
skupa (genitive singular)
skupova (genitive plural)
skupu (dative singular)
skupovima (dative plural)
skup (accusative singular)
skupove (accusative plural)
skupe (singular vocative)
skupovi (plural vocative)
skupu (locative singular)
skupovima (locative plural)
skupom (instrumental singular)
skupovima (instrumental plural)

Deverbative of skȕpiti (“to collect, gather”).

prazan skup


supremum

LL-Q1860 (eng)-Flame, not lame-supremum.wav

A sublattice of a lattice is a subset that is closed under pairwise infima and suprema.

2006, Charalambos D. Aliprantis, Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, 3rd edition, Springer, page 8:

The best way to describe the supremum of S is to say that it wants to be the greatest element of S. In fact, if S has a greatest element, then that element is the supremum.

2010, James S. Howland, Basic Real Analysis, Jones & Bartlett Publishers, page 9:

The key to an approach to vector optimization based on infimum and supremum is to consider set-based objective functions and to extend the partial ordering of the original objective space to a suitable subspace of the power set. In this new space the infimum and supremum exist under the usual assumptions.

2011, Andreas Löhne, Vector Optimization with Infimum and Supremum, Springer, page vii:

partially ordered set
partially ordered set

(set theory) (real analysis): Given a subset X of R, the smallest real number that is ≥ every element of X; (order theory): given a subset X of a partially ordered set P (with partial order ≤), the least element y of P such that every element of X is ≤ y.

上确界

element of a set greater than or equal to all members of a given subset


yhdiste

yhdiste (nominative singular)
yhdisteet (nominative plural)
yhdiste (accusative nominative singular)
yhdisteet (accusative nominative plural)
yhdisteen (accusative genitive singular)
yhdisteet (accusative genitive plural)
yhdisteen (genitive singular)
yhdisteiden (genitive plural)
yhdisteitten (genitive plural)
yhdistettä (partitive singular)
yhdisteitä (partitive plural)
yhdisteessä (inessive singular)
yhdisteissä (inessive plural)
yhdisteestä (elative singular)
yhdisteistä (elative plural)
yhdisteeseen (illative singular)
yhdisteisiin (illative plural)
yhdisteihin (illative plural)
yhdisteellä (adessive singular)
yhdisteillä (adessive plural)
yhdisteeltä (ablative singular)
yhdisteiltä (ablative plural)
yhdisteelle (allative singular)
yhdisteille (allative plural)
yhdisteenä (essive singular)
yhdisteinä (essive plural)
yhdisteeksi (singular translative)
yhdisteiksi (plural translative)
yhdisteettä (abessive singular)
yhdisteittä (abessive plural)
yhdistein (instructive plural)
See the possessive forms below. (comitative)
yhdisteeni (nominative singular)
yhdisteeni (nominative plural)
yhdisteeni (accusative nominative singular)
yhdisteeni (accusative nominative plural)
yhdisteeni (accusative genitive singular)
yhdisteeni (accusative genitive plural)
yhdisteeni (genitive singular)
yhdisteideni (genitive plural)
yhdisteitteni (genitive plural)
yhdistettäni (partitive singular)
yhdisteitäni (partitive plural)
yhdisteessäni (inessive singular)
yhdisteissäni (inessive plural)
yhdisteestäni (elative singular)
yhdisteistäni (elative plural)
yhdisteeseeni (illative singular)
yhdisteisiini (illative plural)
yhdisteihini (illative plural)
yhdisteelläni (adessive singular)
yhdisteilläni (adessive plural)
yhdisteeltäni (ablative singular)
yhdisteiltäni (ablative plural)
yhdisteelleni (allative singular)
yhdisteilleni (allative plural)
yhdisteenäni (essive singular)
yhdisteinäni (essive plural)
yhdisteekseni (singular translative)
yhdisteikseni (plural translative)
yhdisteettäni (abessive singular)
yhdisteittäni (abessive plural)
yhdisteineni (comitative plural)
yhdisteesi (nominative singular)
yhdisteesi (nominative plural)
yhdisteesi (accusative nominative singular)
yhdisteesi (accusative nominative plural)
yhdisteesi (accusative genitive singular)
yhdisteesi (accusative genitive plural)
yhdisteesi (genitive singular)
yhdisteidesi (genitive plural)
yhdisteittesi (genitive plural)
yhdistettäsi (partitive singular)
yhdisteitäsi (partitive plural)
yhdisteessäsi (inessive singular)
yhdisteissäsi (inessive plural)
yhdisteestäsi (elative singular)
yhdisteistäsi (elative plural)
yhdisteeseesi (illative singular)
yhdisteisiisi (illative plural)
yhdisteihisi (illative plural)
yhdisteelläsi (adessive singular)
yhdisteilläsi (adessive plural)
yhdisteeltäsi (ablative singular)
yhdisteiltäsi (ablative plural)
yhdisteellesi (allative singular)
yhdisteillesi (allative plural)
yhdisteenäsi (essive singular)
yhdisteinäsi (essive plural)
yhdisteeksesi (singular translative)
yhdisteiksesi (plural translative)
yhdisteettäsi (abessive singular)
yhdisteittäsi (abessive plural)
yhdisteinesi (comitative plural)
yhdisteemme (nominative singular)
yhdisteemme (nominative plural)
yhdisteemme (accusative nominative singular)
yhdisteemme (accusative nominative plural)
yhdisteemme (accusative genitive singular)
yhdisteemme (accusative genitive plural)
yhdisteemme (genitive singular)
yhdisteidemme (genitive plural)
yhdisteittemme (genitive plural)
yhdistettämme (partitive singular)
yhdisteitämme (partitive plural)
yhdisteessämme (inessive singular)
yhdisteissämme (inessive plural)
yhdisteestämme (elative singular)
yhdisteistämme (elative plural)
yhdisteeseemme (illative singular)
yhdisteisiimme (illative plural)
yhdisteihimme (illative plural)
yhdisteellämme (adessive singular)
yhdisteillämme (adessive plural)
yhdisteeltämme (ablative singular)
yhdisteiltämme (ablative plural)
yhdisteellemme (allative singular)
yhdisteillemme (allative plural)
yhdisteenämme (essive singular)
yhdisteinämme (essive plural)
yhdisteeksemme (singular translative)
yhdisteiksemme (plural translative)
yhdisteettämme (abessive singular)
yhdisteittämme (abessive plural)
yhdisteinemme (comitative plural)
yhdisteenne (nominative singular)
yhdisteenne (nominative plural)
yhdisteenne (accusative nominative singular)
yhdisteenne (accusative nominative plural)
yhdisteenne (accusative genitive singular)
yhdisteenne (accusative genitive plural)
yhdisteenne (genitive singular)
yhdisteidenne (genitive plural)
yhdisteittenne (genitive plural)
yhdistettänne (partitive singular)
yhdisteitänne (partitive plural)
yhdisteessänne (inessive singular)
yhdisteissänne (inessive plural)
yhdisteestänne (elative singular)
yhdisteistänne (elative plural)
yhdisteeseenne (illative singular)
yhdisteisiinne (illative plural)
yhdisteihinne (illative plural)
yhdisteellänne (adessive singular)
yhdisteillänne (adessive plural)
yhdisteeltänne (ablative singular)
yhdisteiltänne (ablative plural)
yhdisteellenne (allative singular)
yhdisteillenne (allative plural)
yhdisteenänne (essive singular)
yhdisteinänne (essive plural)
yhdisteeksenne (singular translative)
yhdisteiksenne (plural translative)
yhdisteettänne (abessive singular)
yhdisteittänne (abessive plural)
yhdisteinenne (comitative plural)
yhdisteensä (nominative singular)
yhdisteensä (nominative plural)
yhdisteensä (accusative nominative singular)
yhdisteensä (accusative nominative plural)
yhdisteensä (accusative genitive singular)
yhdisteensä (accusative genitive plural)
yhdisteensä (genitive singular)
yhdisteidensä (genitive plural)
yhdisteittensä (genitive plural)
yhdistettään (partitive singular)
yhdistettänsä (partitive singular)
yhdisteitään (partitive plural)
yhdisteitänsä (partitive plural)
yhdisteessään (inessive singular)
yhdisteessänsä (inessive singular)
yhdisteissään (inessive plural)
yhdisteissänsä (inessive plural)
yhdisteestään (elative singular)
yhdisteestänsä (elative singular)
yhdisteistään (elative plural)
yhdisteistänsä (elative plural)
yhdisteeseensä (illative singular)
yhdisteisiinsä (illative plural)
yhdisteihinsä (illative plural)
yhdisteellään (adessive singular)
yhdisteellänsä (adessive singular)
yhdisteillään (adessive plural)
yhdisteillänsä (adessive plural)
yhdisteeltään (ablative singular)
yhdisteeltänsä (ablative singular)
yhdisteiltään (ablative plural)
yhdisteiltänsä (ablative plural)
yhdisteelleen (allative singular)
yhdisteellensä (allative singular)
yhdisteilleen (allative plural)
yhdisteillensä (allative plural)
yhdisteenään (essive singular)
yhdisteenänsä (essive singular)
yhdisteinään (essive plural)
yhdisteinänsä (essive plural)
yhdisteekseen (singular translative)
yhdisteeksensä (singular translative)
yhdisteikseen (plural translative)
yhdisteiksensä (plural translative)
yhdisteettään (abessive singular)
yhdisteettänsä (abessive singular)
yhdisteittään (abessive plural)
yhdisteittänsä (abessive plural)
yhdisteineen (comitative plural)
yhdisteinensä (comitative plural)

(set theory) union (set containing all of the elements of two or more sets)


模糊集合

ㄇㄛˊ ㄏㄨˊ ㄐㄧˊ ㄏㄜˊ

ㄇㄛˊ ㄏㄨˊ ㄐㄧˊ ㄏㄜˊ

/mu̯ɔ³⁵ xu³⁵ t͡ɕi³⁵ xɤ³⁵/

/mu̯ɔ³⁵ xu³⁵ t͡ɕi³⁵ xɤ³⁵/


conjunto de partes

pt-noun

conjunto de partes m (plural conjuntos de partes)

(set theory) power set (set of all subsets of a set)


set-builder notation

With this idea for describing a finite set of sets, it is easy to generalize the concept to a certain infinite family #92;mathcal#123;S#125;#95;2 of sets #92;mathcal#123;S#125;#95;2#61;#92;#123;A#95;i#92;verti#92;inN#92;#125;#61;#92;#123;A#95;1,A#95;2,A#95;3,#92;dots,A#95;n,#92;dots#92;#125;. Once again, the power of set builder notation triumphs. The sets #92;mathcal#123;S#125;#95;1 and #92;mathcal#123;S#125;#95;2 may be described more precisely with set builder notation than by enumeration.

2000, Kenneth E. Hummel, Introductory Concepts for Abstract Mathematics, CRC Press (Chapman & Hall/CRC), page 123:

Q=a/b|a∈Iandb∈I,b ne 0
This statement is read in English as "Q is the set of all numbers of the form a/b such that a and b are both integers, but b is not equal to zero."

2011, Tom Bassarear, Mathematics for Elementary School Teachers, Cengage Learning, 5th Edition, page 56, In this case, and in many other cases, we describe the set using set-builder notation

2012, Richard N. Aufmann, Joanne Lockwood, Intermediate Algebra, Cengage Learning, 8th Edition, page 6,
A second method of representing a set is set-builder notation. Set-builder notation can be used to describe almost any set, but it is especially useful when writing infinite sets. In set-builder notation, the set of integers > −3 is written
x|x>-3,x∈integers

(set theory) A mathematical notation for describing a set by specifying the properties that its members must satisfy.


poset

Abbreviation of partially ordered set. Coined by American mathematician Garrett Birkhoff in his Lattice Theory.

coinage

1973, Barbara L. Osofsky, Homological Dimensions of Modules, American Mathematical Society, →ISBN, page 76,
42. Definition. A poset (partially ordered set) (X, ≤) (usually written just X) is a set X together with a transitive, antisymmetric relation ≤ on X.
43. Definition. A linearly ordered set or chain is a poset (X, ≤), such that ∀a, b ∈ X, either a ≤ b or b ≤ a or a = b.

1998, Yuri A. Drozd, Representations of bisected posets and reflection functors, Idun Reiten, Sverre O. Smalø, Øyvind Solberg (editors), Algebras and Modules II, American Mathematical Society (for Canadian Mathematical Society), page 153,
We construct a complete set of reflection functors for the representations of posets and prove that they really have the usual properties. In particular, when the poset is of finite representation type, all of its indecomposable representations can be obtained from some "trivial" ones via relations. To define such reflection functors, a wider class of matrix problem is introduced, called "representations of bisected posets".

The combinatorial interest in posets is largely due to two unoriented graphs associated with a given poset: the comparability graph (which we shall not consider here) and the covering graph.

1999, Manfred Stern, Semimodular Lattices: Theory and Applications, Cambridge University Press, page 189:

(set theory, order theory) A partially ordered set.


joukko

joukko (nominative singular)
joukot (nominative plural)
joukko (accusative nominative singular)
joukot (accusative nominative plural)
joukon (accusative genitive singular)
joukot (accusative genitive plural)
joukon (genitive singular)
joukkojen (genitive plural)
joukkoa (partitive singular)
joukkoja (partitive plural)
joukossa (inessive singular)
joukoissa (inessive plural)
joukosta (elative singular)
joukoista (elative plural)
joukkoon (illative singular)
joukkoihin (illative plural)
joukolla (adessive singular)
joukoilla (adessive plural)
joukolta (ablative singular)
joukoilta (ablative plural)
joukolle (allative singular)
joukoille (allative plural)
joukkona (essive singular)
joukkoina (essive plural)
joukoksi (singular translative)
joukoiksi (plural translative)
joukotta (abessive singular)
joukoitta (abessive plural)
joukoin (instructive plural)
See the possessive forms below. (comitative)
joukkoni (nominative singular)
joukkoni (nominative plural)
joukkoni (accusative nominative singular)
joukkoni (accusative nominative plural)
joukkoni (accusative genitive singular)
joukkoni (accusative genitive plural)
joukkoni (genitive singular)
joukkojeni (genitive plural)
joukkoani (partitive singular)
joukkojani (partitive plural)
joukossani (inessive singular)
joukoissani (inessive plural)
joukostani (elative singular)
joukoistani (elative plural)
joukkooni (illative singular)
joukkoihini (illative plural)
joukollani (adessive singular)
joukoillani (adessive plural)
joukoltani (ablative singular)
joukoiltani (ablative plural)
joukolleni (allative singular)
joukoilleni (allative plural)
joukkonani (essive singular)
joukkoinani (essive plural)
joukokseni (singular translative)
joukoikseni (plural translative)
joukottani (abessive singular)
joukoittani (abessive plural)
joukkoineni (comitative plural)
joukkosi (nominative singular)
joukkosi (nominative plural)
joukkosi (accusative nominative singular)
joukkosi (accusative nominative plural)
joukkosi (accusative genitive singular)
joukkosi (accusative genitive plural)
joukkosi (genitive singular)
joukkojesi (genitive plural)
joukkoasi (partitive singular)
joukkojasi (partitive plural)
joukossasi (inessive singular)
joukoissasi (inessive plural)
joukostasi (elative singular)
joukoistasi (elative plural)
joukkoosi (illative singular)
joukkoihisi (illative plural)
joukollasi (adessive singular)
joukoillasi (adessive plural)
joukoltasi (ablative singular)
joukoiltasi (ablative plural)
joukollesi (allative singular)
joukoillesi (allative plural)
joukkonasi (essive singular)
joukkoinasi (essive plural)
joukoksesi (singular translative)
joukoiksesi (plural translative)
joukottasi (abessive singular)
joukoittasi (abessive plural)
joukkoinesi (comitative plural)
joukkomme (nominative singular)
joukkomme (nominative plural)
joukkomme (accusative nominative singular)
joukkomme (accusative nominative plural)
joukkomme (accusative genitive singular)
joukkomme (accusative genitive plural)
joukkomme (genitive singular)
joukkojemme (genitive plural)
joukkoamme (partitive singular)
joukkojamme (partitive plural)
joukossamme (inessive singular)
joukoissamme (inessive plural)
joukostamme (elative singular)
joukoistamme (elative plural)
joukkoomme (illative singular)
joukkoihimme (illative plural)
joukollamme (adessive singular)
joukoillamme (adessive plural)
joukoltamme (ablative singular)
joukoiltamme (ablative plural)
joukollemme (allative singular)
joukoillemme (allative plural)
joukkonamme (essive singular)
joukkoinamme (essive plural)
joukoksemme (singular translative)
joukoiksemme (plural translative)
joukottamme (abessive singular)
joukoittamme (abessive plural)
joukkoinemme (comitative plural)
joukkonne (nominative singular)
joukkonne (nominative plural)
joukkonne (accusative nominative singular)
joukkonne (accusative nominative plural)
joukkonne (accusative genitive singular)
joukkonne (accusative genitive plural)
joukkonne (genitive singular)
joukkojenne (genitive plural)
joukkoanne (partitive singular)
joukkojanne (partitive plural)
joukossanne (inessive singular)
joukoissanne (inessive plural)
joukostanne (elative singular)
joukoistanne (elative plural)
joukkoonne (illative singular)
joukkoihinne (illative plural)
joukollanne (adessive singular)
joukoillanne (adessive plural)
joukoltanne (ablative singular)
joukoiltanne (ablative plural)
joukollenne (allative singular)
joukoillenne (allative plural)
joukkonanne (essive singular)
joukkoinanne (essive plural)
joukoksenne (singular translative)
joukoiksenne (plural translative)
joukottanne (abessive singular)
joukoittanne (abessive plural)
joukkoinenne (comitative plural)
joukkonsa (nominative singular)
joukkonsa (nominative plural)
joukkonsa (accusative nominative singular)
joukkonsa (accusative nominative plural)
joukkonsa (accusative genitive singular)
joukkonsa (accusative genitive plural)
joukkonsa (genitive singular)
joukkojensa (genitive plural)
joukkoaan (partitive singular)
joukkoansa (partitive singular)
joukkojaan (partitive plural)
joukkojansa (partitive plural)
joukossaan (inessive singular)
joukossansa (inessive singular)
joukoissaan (inessive plural)
joukoissansa (inessive plural)
joukostaan (elative singular)
joukostansa (elative singular)
joukoistaan (elative plural)
joukoistansa (elative plural)
joukkoonsa (illative singular)
joukkoihinsa (illative plural)
joukollaan (adessive singular)
joukollansa (adessive singular)
joukoillaan (adessive plural)
joukoillansa (adessive plural)
joukoltaan (ablative singular)
joukoltansa (ablative singular)
joukoiltaan (ablative plural)
joukoiltansa (ablative plural)
joukolleen (allative singular)
joukollensa (allative singular)
joukoilleen (allative plural)
joukoillensa (allative plural)
joukkonaan (essive singular)
joukkonansa (essive singular)
joukkoinaan (essive plural)
joukkoinansa (essive plural)
joukokseen (singular translative)
joukoksensa (singular translative)
joukoikseen (plural translative)
joukoiksensa (plural translative)
joukottaan (abessive singular)
joukottansa (abessive singular)
joukoittaan (abessive plural)
joukoittansa (abessive plural)
joukkoineen (comitative plural)
joukkoinensa (comitative plural)

From Proto-Finnic *joukko, borrowed from Proto-Germanic *jaukiją or from older Proto-Indo-European *yewg- (see the Proto-Finnic entry for more).

Varsinaisessa joukko-opissa joukko voi sisältää vain toisia joukkoja.

In proper set theory, a set may only contain other sets.

(mathematics, set theory) set (collection of objects, or of other sets)


vereniging

nl-noun

vereniging f (plural verenigingen, diminutive vereniginkje n)

verenigingen (plural)
vereniginkje (diminutive neuter)

Negerhollands: vereeniging

→ Peranakan Indonesian: vereeniging


集合

/t͡ɕi³⁵ xɤ³⁵/

/t͡siʔ⁴⁻²¹ haʔ⁴/

/t͡siʔ⁴⁻² haʔ⁴/

/t͡sip̚²⁴⁻² hap̚²⁴/

/t͡sip̚¹²¹⁻²¹ hap̚¹²¹/

/t͡ɕi³⁵ xɤ³⁵/

/t͡siʔ⁴⁻²¹ haʔ⁴/

/t͡siʔ⁴⁻² haʔ⁴/

/t͡sip̚²⁴⁻² hap̚²⁴/

/t͡sip̚¹²¹⁻²¹ hap̚¹²¹/


ZFC

Zermelo-Fraenkel set theory with Choice
w:Zermelo–Fraenkel set theory

(set theory) Initialism of Zermelo-Fraenkel set theory with Choice; the standard axiomatization of set theory, including the axiom of choice.


axiom of choice

en-noun

axiom of choice (plural axioms of choice)

A calque of German Axiom der Auswahl (now more commonly Auswahlaxiom), which first appeared in print with a description of the axiom in 1908, Ernst Zermelo, Untersuchungen über die Grundlagen der Mengenlehre I ["Investigations in the foundations of set theory I"], Mathematische Annalen, 65 (although the paper was dated 1907).

The axiom of choice is logically equivalent to the assertion that every vector space has a basis.

If V = L then the axioms of choice and the continuum hypothesis are both true, and the assertion that a measurable cardinal exists is false.

1993, Thomas Tymoczko, editor, Penelope Maddy: Does V Equal L?: New Directions in the Philosophy of Mathematics: An Anthology, page 357:

Throughout this work we adopt the Zermelo–Fraenkel (ZF) axioms of set theory with the Axiom of Choice, commonly abbreviated as ZFC. It follows from the work of Gödel and Cohen that if the ZF axioms are consistent, the Axiom of Choice can be neither proved nor disproved from the ZF axioms.

1993, Gary L. Wise, Eric B. Hall, Counterexamples in Probability and Real Analysis, page vii:

To clarify these ideas for the reader, let us show, without the axiom of choice, that a product of finitely many nonempty sets is nonempty: This is done by induction on the number n of sets.[…]The finite axiom of choice is not an axiom, but rather a theorem that can be proved from the other axioms. In contrast, there are weak forms of the axiom of choice that are not provable.

2000, Bruno Poizat, translated by Moses Klein, A Course in Model Theory: An Introduction to Contemporary Mathematical Logic, page 169:

Perhaps what this debate about whether to accept the axiom of choice indicates is that the disjunction between regularity and randomness is as fundamental to our conception of the world as that between discreteness and continuity.

2004, Michael Potter, Set Theory and its Philosophy: A Critical Introduction, page 259:

There is a uniform status quo almost with just a few exceptions of a handful of mathematicians around the world. Most people accept the “infinite choice, infinite decimals” approach to real numbers, and the justification that has been created to substantiate this, and to overcome some of the difficulties that I have shown you is through an elaborate axiomatic framework. Mathematicians are not stupid! They realize that this theory is dubious, and so what they have done is they have created a rather elaborate axiomatic framework. And that axiomatic framework is something that we can study in set theory and logic; and its purported aim is to create a framework for mathematics; but its initial aims were very much directed towards solving the problems, overcoming the difficulties with real numbers as infinite decimals. Prominent amongst these axioms is this Axiom of Choice! The Axiom of Choice manifests itself in mathematics in many ways; but its primary role — OK — its primary role is exactly here at the level of infinite decimals and real numbers. It essentially asserts, as a matter of faith or belief, that it is possible to choose an infinite number of digits arbitrarily and independently, and that a legitimate mathematical object results. This is the key philosophical point. Are we, or are we not, in a position to be able to specify a first digit, a second digit, a third digit, a fourth digit, and so on to infinity? And are we allowed to call that specification a new mathematical object, a real number? The Axiom of Choice says that “Yes we are!”, so it is a statement of belief and its primary objective is to allow the building of this real number system.

2012, 47:31 – 49:50 from the start, in Difficulties with real numbers as infinite decimals (I) Real numbers + limits Math Foundations 91, episode 91, njwildberger (N. J. Wildberger), via YouTube:

(set theory) One of the axioms of set theory, equivalent to the statement that an arbitrary direct product of non-empty sets is non-empty; any version of said axiom, for example specifying the cardinality of the number of sets from which choices are made.

ընտրության աքսիոմ

axiom that any product of non-empty sets is non-empty


direct product

suora tulo

product of sets — see also Cartesian product

If A and B are sets, their direct product is the set of ordered pairs (a,b) with a in A and b in B.

(set theory) The set of all possible tuples whose elements are elements of given, separately specified, sets.