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set: Definition, Synonyms and Much More from Answers.com

️Wed Jul 01 2015

This article is about mathematical sets. For other uses, see Set (disambiguation).

This article gives a basic introduction to what mathematicians call "intuitive" or "naive" set theory; for a more detailed account see Naive set theory. For a rigorous modern axiomatic treatment of sets, see Axiomatic set theory.

In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. Although this appears to be a simple idea, sets are one of the most fundamental concepts in modern mathematics. The study of the structure of possible sets, set theory, is rich and ongoing. Having only been invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics education, being introduced from primary school in many countries. Set theory can be viewed as the foundation upon which nearly all of mathematics can be derived.

The intersection of two sets is made up of the objects contained in both sets, shown in a Venn diagram.

At the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre, Georg Cantor, the principal creator of set theory, gave the following definition of a set:^[1]

By a "set" we mean any collection M into a whole of definite, distinct objects m (which are called the "elements" of M) of our perception [Anschauung] or of our thought.

The elements of a set, also called its members, can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with capital letters. The statement that sets A and B are equal means that they have precisely the same members (i.e., every member of A is also a member of B and vice versa).

Unlike a multiset, every element of a set must be unique; no two members may be identical. All set operations preserve the property that each element of the set is unique. The order in which the elements of a set are listed is irrelevant, unlike a sequence or tuple.

Describing sets

There are two ways of describing, or specifying the members of, a set. One way is by intensional definition, using a rule or semantic description. See this example:

A is the set whose members are the first four positive integers.

B is the set of colors of the French flag.

The second way is by extension, that is, listing each member of the set. An extensional definition is notated by enclosing the list of members in braces:

C = {4, 2, 1, 3}

D = {blue, white, red}

The order in which the elements of a set are listed in an extensional definition is irrelevant, as are any repetitions in the list. For example,

{6, 11} = {11, 6} = {11, 11, 6, 11}

are equivalent, because the extensional specification means merely that each of the elements listed is a member of the set.

For sets with many elements, the enumeration of members can be abbreviated. For instance, the set of the first thousand positive whole numbers may be specified extensionally as:

{1, 2, 3, ..., 1000},

where the ellipsis ("...") indicates that the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members. Thus the set of positive even numbers can be written as {2, 4, 6, 8, ... }.

The notation with braces may also be used in an intensional specification of a set. In this usage, the braces have the meaning "the set of all ..." So E = {playing-card suits} is the set whose four members are ♠, ♦, ♥, and ♣. A more general form of this is set-builder notation, through which, for instance, the set F of the twenty smallest integers that are four less than perfect squares can be denoted:

F = {n² – 4 : n is an integer; and 0 ≤ n ≤ 19}

In this notation, the colon (":") means "such that", and the description can be interpreted as "F is the set of all numbers of the form n² – 4, such that n is a whole number in the range from 0 to 19 inclusive." Sometimes the vertical bar ("|") is used instead of the colon.

One often has the choice of specifying a set intensionally or extensionally. In the examples above, for instance, A = C and B = D.

Membership

If something is or is not an element of a particular set then this is symbolised by $\in$ and $\notin$ respectively. So, with respect to the sets defined above:

Cardinality

The cardinality |S| of a set S is "the number of members of S." For example, since the French flag has three colors, |B| = 3.

There is a set with no members and zero cardinality, which is called the empty set (or the null set) and is denoted by the symbol ø. For example, the set A of all three-sided squares has zero members (|A| = 0), and thus A = ø. Though, like the number zero, it may seem trivial, the empty set is quite important in mathematics. The existence of this set is one of the fundamental concepts of axiomatic set theory.

Some sets have infinite cardinality. The set N of natural numbers, for instance, is infinite. Some infinite cardinalities are greater than others. For instance, the set of real numbers has greater cardinality than the set of natural numbers. However, it can be shown that the cardinality of (which is to say, the number of points on) a straight line is the same as the cardinality of any segment of that line, of an entire plane, and indeed of any Euclidean space.

Subsets

If every member of set A is also a member of set B, then A is said to be a subset of B, written $A \subseteq B$ (also pronounced A is contained in B). Equivalently, we can write $B \supseteq A$ , read as B is a superset of A, B includes A, or B contains A. The relationship between sets established by $\subseteq$ is called inclusion or containment.

If A is a subset of, but not equal to, B, then A is called a proper subset of B, written $A \subsetneq B$ (A is a proper subset of B) or $B \supsetneq A$ (B is proper superset of A).

Note that the expressions $A\subset B$ and $A\supset B$ are used differently by different authors; some authors use them to mean the same as $A\subseteq B$ (respectively $A\supseteq B$ ), whereas other use them to mean the same as $A\subsetneq B$ (respectively $A\supsetneq B$ ).

A is a subset of B

Example:

The empty set is a subset of every set and every set is a subset of itself:

$\emptyset \subseteq A$
$A \subseteq A$

Power set

The power set of a set S can be defined as the set of all subsets of S. This includes the subsets formed from the members of S and the empty set. If a finite set S has cardinality n then the power set of S has cardinality 2ⁿ. If S is an infinite (either countable or uncountable) set then the power set of S is always uncountable. The power set can be written as 2^S.

As an example, the power set 2^{{1, 2, 3}} of {1, 2, 3} is equal to the set {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ø}. The cardinality of the original set is 3, and that of the power set is eight, which is equal to two to the third. This relationship is one of the reasons for the terminology power set. Similarly, its notation is an example of a general convention providing notations for sets based on their cardinalities.

Special sets

There are some sets which hold great mathematical importance and are referred to with such regularity that they have acquired special names and notational conventions to identify them. One of these is the empty set. Many of these sets are represented using Blackboard bold typeface. Special sets of numbers include:

$\mathbb{P}$ , denoting the set of all primes.
$\mathbb{N}$ , denoting the set of all natural numbers. That is to say, $\mathbb{N}$ = {1, 2, 3, ...}, or sometimes $\mathbb{N}$ = {0, 1, 2, 3, ...}.
$\mathbb{Z}$ , denoting the set of all integers (whether positive, negative or zero). So $\mathbb{Z}$ = {..., -2, -1, 0, 1, 2, ...}.
$\mathbb{Q}$ , denoting the set of all rational numbers (that is, the set of all proper and improper fractions). So, $\mathbb{Q} = \left\{ \begin{matrix}\frac{a}{b} \end{matrix}: a,b \in \mathbb{Z}, b \neq 0\right\}$ . For example, $\begin{matrix} \frac{1}{4} \end{matrix} \in \mathbb{Q}$ and $\begin{matrix}\frac{11}{6} \end{matrix} \in \mathbb{Q}$ . All integers are in this set since every integer a can be expressed as the fraction $\begin{matrix} \frac{a}{1} \end{matrix}$ .
$\mathbb{R}$ , denoting the set of all real numbers. This set includes all rational numbers, together with all irrational numbers (that is, numbers which cannot be rewritten as fractions, such as π, e, and √2).
$\mathbb{C}$ , denoting the set of all complex numbers.

Each of these sets of numbers has an infinite number of elements, and $\mathbb{P} \subset \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$ . The primes are used less frequently than the others outside of number theory and related fields.

Basic operations

Unions

There are ways to construct new sets from existing ones. Two sets can be "added" together. The union of A and B, denoted by A U B, is the set of all things which are members of either A or B.

The union of A and B

Examples:

{1, 2} U {red, white} = {1, 2, red, white}
{1, 2, green} U {red, white, green} = {1, 2, red, white, green}
{1, 2} U {1, 2} = {1, 2}

Some basic properties of unions are:

A U B = B U A
A ⊆ A U B
A U A = A
A U ø = A
A ⊆ B if and only if A U B = B

Intersections

A new set can also be constructed by determining which members two sets have "in common". The intersection of A and B, denoted by A ∩ B, is the set of all things which are members of both A and B. If A ∩ B = ø, then A and B are said to be disjoint.

The intersection of A and B

Examples:

{1, 2} ∩ {red, white} = ø
{1, 2, green} ∩ {red, white, green} = {green}
{1, 2} ∩ {1, 2} = {1, 2}

Some basic properties of intersections:

A ∩ B = B ∩ A
A ∩ B ⊆ A
A ∩ A = A
A ∩ ø = ø
A ⊆ B if and only if A ∩ B = A

Complements

Two sets can also be "subtracted". The relative complement of A in B (also called the set theoretic difference of B and A), denoted by B \ A, (or B − A) is the set of all elements which are members of B, but not members of A. Note that it is valid to "subtract" members of a set that are not in the set, such as removing green from {1,2,3}; doing so has no effect.

In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \ A, is called the absolute complement or simply complement of A, and is denoted by A′.

The relative complement
of A in B

The complement of A in U

Examples:

{1, 2} \ {red, white} = {1, 2}
{1, 2, green} \ {red, white, green} = {1, 2}
{1, 2} \ {1, 2} = ø
If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then the complement of E in U is O, or equivalently, E′ = O.

Some basic properties of complements:

A U A′ = U
A ∩ A′ = ø
(A′ )′ = A
A \ A = ø
A \ B = A ∩ B′

Cartesian product

A new set can be constructed by combining every element of one set with every element of a different set. The cartesian product of two sets A and B, denoted by A × B is the set of all ordered pairs (a,b) such that a is a member of A and b is a member of B.

Examples:

{1, 2} × {red, white} = {(1,red), (1,white), (2,red), (2,white)}
{1, 2, green} × {red, white, green} = {(1,red), (1,white), (1,green), (2,red), (2,white), (2,green), (green,red), (green,white), (green,green)}
{1, 2} × {1, 2} = {(1,1), (1,2), (2,1), (2,2)}

Some basic properties of cartesian products:

A × ø = ø
A × (B U C) = (A × B) U (A × C)
|A × B| = |A| x |B|

Applications

Set theory is seen as the foundation from which virtually all of mathematics can be derived. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.

One of the main applications of naive set theory is constructing relations. A relation from a domain A to a codomain B is nothing but a subset of A × B. Given this concept, we are quick to see that the set F of all ordered pairs (x, x²), where x is real, is quite familiar. It has a domain set $\mathbb{R}$ and a codomain set that is also $\mathbb{R}$ , because the set of all squares is subset of the set of all reals. If placed in functional notation, this relation becomes f( x ) = x². The reason these two are equivalent is for any given value, y that the function is defined for, it's corresponding ordered pair, (y, y²) is a member of the set F.

Axiomatic set theory

Although initially the naive set theory, which defines a set merely as any well-defined collection, was well accepted, it soon ran into several obstacles. It was found that this definition spawned several paradoxes, most notably:

Russell's paradox - It shows that the "set of all sets which do not contain themselves," i.e. the "set" $\left \{ x: x\mbox{ is a set and }x\notin x \right \}$ does not exist.
Cantor's paradox - It shows that "the set of all sets" cannot exist.

The reason is that the phrase well-defined is not very well-defined. It was important to free set theory of these paradoxes because nearly all of mathematics was being redefined in terms of set theory. In an attempt to avoid these paradoxes, set theory was axiomatized based on first-order logic, and thus the axiomatic set theory was born.

For most purposes however, the naive set theory is still useful.

Notes

^ Quoted in Dauben, p. 170.

References

Dauben, Joseph W., Georg Cantor: His Mathematics and Philosophy of the Infinite, Boston: Harvard University Press (1979) ISBN 978-0-691-02447-9.
Halmos, Paul R., Naive Set Theory, Princeton, N.J.: Van Nostrand (1960) ISBN 0-387-90092-6.
Stoll, Robert R., Set Theory and Logic, Mineola, N.Y.: Dover Publications (1979) ISBN 0-486-63829-4.

External links

Wikimedia Commons has media related to:

C2 Wiki - Examples of set operations using English operators.

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be-x-old:Мноства zh-classical:集 nov:Ensemble

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Dansk (Danish)
1.
v. tr. - sætte, stille, lægge, indstille, give, beramme, indfatte
v. intr. - gå ned, stivne, størkne, afbinde

idioms:

be set in one's ways have faste vaner
set about gå i gang med, begynde på
set against sammenligne
set back koste, forsinke
set by sætte til side, henlægge
set down sætte ned, sætte af, indføre, foreskrive, skrive ned
set down as anse
set eyes on se, se for sine øjne
set great store by lægge vægt på, sætte pris på
set in begynde for alvor, sætte ind
set in concrete ligge helt fast
set off starte, tage af sted, udløse, sætte i gang, affyre, udligne, modregne, sætte i relief
set on angribe, overfalde, pudse på
set one's house in order få orden på sit hus
set oneself up as udgive sig for at være, optræde som, etablere sig som
set out tage af sted, tage ud, fremsætte, fremføre, afstikke, lægge frem
set out for tage af sted til
set the record straight få tingene på det rene, opklare en misforståelse
set up opsætte, rejse, opstille, stifte, etablere, indføre, udstøde
set upon angribe, overfalde

2.
n. - sæt

3.
adj. - fast, fastsat, foreskreven, bestemt

Nederlands (Dutch)
zetten, plaatsen, situeren, watergolven, hard worden, stel, decor, set, verzameling, groep, kliek, pakket, klaar, bepaald, vastbesloten

Français (French)
1.
v. tr. - placer, poster, sertir, mettre (une table), tendre (un piège), fixer (une date), établir (un record), mettre (qch) à l'heure, mettre (qch) en marche, donner (des devoirs), poser (un problème), préparer (un examen), (Cin, Théât, Littérat, TV) situer, (Mus) mettre en musique, (Imprim) composer, (Méd) immobiliser, éclisser, faire une mise en plis, faire prendre/durcir (le béton), placer/estimer qn (au-dessus, en dessous de qn), (GB, École) grouper (qn) par niveau
v. intr. - se coucher (le soleil), prendre/durcir (le béton), sécher (la colle), (Méd) se ressouder

idioms:

be set in one's ways avoir ses habitudes
set about se mettre à, commencer à, attaquer (qn), faire courir (une rumeur)
set against dresser contre, monter (qn) contre, s'opposer à qch, confronter qch à qch
set at se mettre à
set back reculer, retarder, être en retrait, coûter les yeux de la tête à
set by mettre de côté
set down poser, déposer, fixer, consigner par écrit, enregistrer, poser (un hélicoptère)
set down as enregistrer comme
set eyes on choisir des yeux
set forward mettre en avant
set great store by accorder de l'importance à
set in commencer, se déclarer, survenir, arriver, s'installer (un ressentiment), (Cout) rapporter
set in concrete coulé dans le béton, intégré au béton
set off partir, déclencher, faire exploser, faire ressortir, (Fin) déduire, faire pleurer (un bébé)
set off against (Fin) déduire (qch) de
set on attaquer qn, lâcher (qch) contre qn
set one's house in order mettre de l'ordre dans ses affaires, ranger sa maison
set oneself up as s'établir à titre de
set out se mettre en route (pour), partir en
set out to avoir pour but de, chercher à, commencer à
set someone against monter (qn) contre
set someone apart prendre (qn) à part
set someone down déposer qn
set someone up aider qn à s'installer, remettre (qn) sur pied, tendre un piège à (un criminel), monter un coup contre qn
set something up monter (un stand), assembler (un équipement), déplier (une chaise), ériger (un barrage routier), dresser (une statue)
set the record straight (fig) mettre les choses au point
set to s'y mettre
set upon attaquer (qn)

2.
n. - jeu (de clés, etc), service (de table), collection (de livres), jeu (d'échecs), paire (de draps), (Sport) set (tennis), (Math) ensemble

3.
adj. - fixé/arrangé (d'avance)

Deutsch (German)
1.
v. - stellen, setzen, vorschreiben, legen, einrichten, zusammenstellen, festsetzen, besetzen, einlegen, decken, einrenken, hart werden

idioms:

be set in one's ways in seinen Gewohnheiten festgefahren sein
set about herfallen über
set against antreiben gegen
set at schätzen, taxieren
set back behindern, zurücksetzen, zurückwerfen
set by zurücklegen, außer acht lassen, aufschieben
set down absetzen, niederschreiben, zuschreiben
set down as halten für
set eyes on jmdn. erblicken
set forward voranbringen
set great store by großen Wert legen auf
set in einsetzen
set in concrete einzementieren
set off auslösen, losfahren
set off against etw. einer Sache (Dat) gegenüberstellen, etw. als Ausgleich für etw. nehmen
set on überfallen
set one's house in order seine Angelegenheiten in Ordnung bringen
set oneself up as sich niederlassen als
set out aufbrechen, aufstellen, beabsichtigen
set out to sich (Dat) vornehmen, etw. zu tun
set someone against jmdn. gegen jmdn. aufbringen
set someone apart sich abheben, auszeichnen
set someone down absetzen, aussteigen
set someone up sich etablieren als
set something up etwas irgendwohin stellen, aufstellen, Anspruch erheben, verursachen
set the record straight keine Mißverständnisse aufkommen lassen
set to sich daranmachen, es sich schmecken lassen, loslegen
set upon attackieren

2.
n. - Satz, Reihe, Setzling, Kreis, Sitz, Bau, Bühnenbild, Szenenaufbau, Gerät, Service, Set, Menge

3.
adj. - fertig, fest, vorgeschrieben, festgesetzt, bestimmt, entschieden, stur

Ελληνική (Greek)
v. - βάζω, θέτω, τοποθετώ, απιθώνω, παραθέτω, στήνω, ορίζω, διορθώνω, ρυθμίζω, σιάζω, τακτοποιώ, στοιχειοθετώ, μελοποιώ, δύω, βασιλεύω, πήζω, δένω (πολύτιμο λίθο), (για ενδύματα) εφαρμόζω
n. - σύνολο, σειρά, σετ, διάταξη, συσκευή, σκηνικό, πλατό, τάση (κοινής γνώμης), (στο τένις) σετ, (μαθημ.) σύνολο, μιζανπλί, ψυχική διάθεση
adj. - (καθ)ορισμένος, σταθερός, τακτός, κατασταλαγμένος, στερεότυπος, αμετακίνητος, έτοιμος, πηγμένος, απλανής

idioms:

be set in one's ways είμαι αμετακίνητος στις συνήθειές μου
set about αποδύομαι σε, καταπιάνομαι με, επιτίθεμαι σε
set against αντιπαραβάλλω με/προς, στρέφω/-ομαι κατά
set back βάζω πίσω (ρολόι), ρίχνω πίσω, κοστίζω, επιβαρύνω, απωθώ, αναχαιτίζω
set by βάζω στην μπάντα
set down απιθώνω, αποβιβάζω, καταγράφω, καταλογίζω
set down as αποδίδω, καταλογίζω, χαρακτηρίζω
set eyes on (πρωτο)βλέπω
set great store by αποδίδω μεγάλη σημασία σε
set in εγκαθίσταμαι, εδραιώνομαι, κατασταλάζω, επικρατώ, παγιώνομαι, (για εποχή του έτους) μπαίνω για τα καλά, (για σκοτάδι κ.λπ.) πέφτω (για τα καλά), (για ψύχος κ.λπ.) δυναμώνω (για τα καλά), (για καιρικές συνθήκες) δείχνω ότι θα διαρκέσω
set in concrete μέσα σε μπετόν
set off ξεκινώ, θέτω σε κίνηση, τονίζω, προβάλλω, αντισταθμίζω, πυροδοτώ, αντιπαραθέτω
set on επιτίθεμαι κατά, παρακινώ σε, βάζω στα ίχνη
set one's house in order τακτοποιώ τα του οίκου μου
set oneself up as αυτοτιτλοφορούμαι (ως)
set out ξεκινώ, εκθέτω, παραθέτω, διατυπώνω, διατάσσω, (προ)διαγράφω
set out for ξεκινώ για
set the record straight επανορθώνω σφάλμα
set up ορθώνω, στήνω, ιδρύω, εγκαθιστώ, ξεσηκώνω, αρχίζω, οργανώνω, εξοπλίζω, ξεσηκώνω, διατυπώνω (θεωρία κ.λπ.), παραθέτω, ενοχοποιώ
set upon επιτίθεμαι εναντίον

Italiano (Italian)
collocare, sistemare, temperare, aggiustare, ridurre, indurirsi, coppia, cerchia, ambiente, scenario, set, pacco, insieme, pronto, fisso, deciso

idioms:

be set in one's ways essere abitudinario
set about accingersi
set against aizzare contro
set apart separare, contraddistinguere
set aside metter da parte
set at ease mettersi comodo
set back impedire, fare indietreggiare
set by mettere da parte
set down metter giù
set down as considerare come
set forth esporre, avviarsi, sottolineare
set in cominciare
set off partire, far esplodere, dare il via a, scatenare
set on aizzare
set oneself up as farsi passare per
set out partire
set out for dirigersi verso
set piece opera artistica convenzionale
set right riaggiustare
set square squadra
set the scene/stage preparare le condizioni per
set up erigere, montare, incastrare
set upon aggredire

Português (Portuguese)
v. - colocar, dispor, regular, fixar
n. - conjunto de (m), grupo (m)

idioms:

be set in one's ways estar acostumado a fazer sempre o mesmo
set back colocar-se atrás, atrasar
set by meter de parte, economizar
set down depor, colocar em baixo
set down as considerar
set in começar
set off fazer iniciar
set on avançar, atacar
set oneself up as iniciar seu próprio negócio
set out iniciar, partir
set out for partir para
set up iniciar uma profissão
set upon agredir

Русский (Russian)
класть, помещаться, сажать, направлять, подготавливать, определять, садиться, комплект, серия, группа, прибор, конфигурация, направленность, неподвижный, установленный

idioms:

be set in one's ways никогда не изменять своим привычкам
set back помещать в дальнем конце, отодвигать, препятствовать
set by приберегать, ценить, уважать
set down письменно излагать, записывать, высаживать(пассажира), ставить (кого-л.) на место, давать отпор, резкий отказ, нагоняй, поездка в один конец
set down as считать, рассматривать
set in начинаться, наступать, устанавливаться, двигаться (в каком-л направлении), сажать (растения), вшивать (рукав), вставка, вставной, встроенный
set off отправляться, выделять, начинать возбуждать, взрывать, выгодно подчеркивать, компенсировать, засчитывать
set on двигаться вперед, приводить в движение, подвергнуться нападению
set oneself up as выдавать себя за кого-л., претендовать на что-л.
set out ставить, выставлять (на продажу), расставлять, накрывать (на стол), уставлять, намереваться, излагать, украшать
set out for отправляться (в путешествие)
set up помещать, ставить, вывешивать, воздвигать, основывать, устанавливать, открывать (какое-л. дело), помочь устроиться, предлагать, формулировать, планировать, поднимать (крик), обеспечивать, восстанавливать (силы), тренировать, причинять, выдавать себя за кого-л., проявлять удовлетворение
set upon двигаться вперед, приводить в движение, подвергнуться нападению

Español (Spanish)
1.
v. tr. - poner, colocar, situar, ubicar, marcar, ondular, endurecer, cuajar, fraguar, poner en hora, reducir, encajar, montar, ajustar, sentar, asentar, poner a empollar, dirigir, destinar, fijar, señalar, plantar, erigir, preparar, alistar, llevar, azuzar, enemistar, adornar, sembrar, engarzar, dar, estimar, apreciar, inmovilizar, sujetar, apretar, (comp) configurar
v. intr. - fraguar, endurecerse, solidificarse, cuajarse, empollar huevos, caer bien, sentar, ponerse, declinar, acabar, apostar, fluir, tender, inclinarse, dedicarse, fijarse (un color), cambiar, deformarse

idioms:

be set in one's ways determinado, pertinaz
set about emprender, ponerse a, difundir, propagar, atacar
set against instigar contra, enemistar con, comparar con, oponerse
set at intento de ganar el afecto de alguien
set back echar atrás, hacer retroceder, atrasar, retrasar, costar, detener, frenar
set by guardar para uso futuro, ahorrar
set down dejar o poner en el suelo, establecer, poner por escrito, apuntar, dejar bajar, dejar apearse, atribuir, hacer aterrizar, considerar, tomar por, fijar, prever
set down as explicarse como, considerar algo como
set eyes on ver, poner el ojo en
set forward poner los ojos encima, ver, mirar
set great store by atribuir importancia a, dar importancia a
set in comenzar, aparecer, surgir
set in concrete definitivo, reforzado con hormigón
set off hacer resaltar, realzar, adornar, embellecer, volar, compensar, salir corriendo, hacer hablar
set off against enemistar con, poner en contra
set on atacar, instigar contra, azuzar, seguir adelante
set one's house in order cuidarse de los asuntos propios
set oneself up as dárselas de, presumir de, meterse a
set out salir, partir, ponerse en camino
set out to empezar a
set someone against indisponer a alguien, malquistar con alguien, oponerse a alguien
set someone apart apartar a alguien
set someone down detener el coche y permitir a alguien bajar
set someone up establecer a alguien en un puesto, sanar a alguien, incriminar dolosamente
set something up erguir, fundar, construir, montar
set the record straight dejar las cosas bien claras, deshacer un error
set to ponerse a
set upon atacar, emprender, instigar contra

2.
n. - pareja, par, círculo de personas, círculo, juego, conjunto

3.
adj. - decorado, determinado, fijado, fijado u organizado con antelación

Svenska (Swedish)
v. - sätta, ställa, lägga, sätta fram, sjunka, gå ner, strömma
n. - sats, uppsättning, set, saker, ställ, garnityr, servis
adj. - fastställd, fast, bestämd

中文（简体） (Chinese (Simplified))
1. 放, 置, 使接触, 竖立, 使处于, 落, 下沉, 凝结, 凝固, 衰落, 固定, 定型, 固定的, 坚决的, 规定的, 一套, 一批, 一副

idioms:

be set in one's ways 固执
set about 着手, 开始
set against 反对, 使对立, 使平衡, 从...扣除, 比较
set back 使受挫折
set by 把...留开
set down 放下, 制定
set down as 认为..., 把...看作
set eyes on 看到, 望见
set great store by 很相信
set in 开始, 上涨, 到来
set in concrete 固定
set off 出发, 使爆炸, 动身
set on 攻击, 怂恿, 前进
set one's house in order 整理
set oneself up as 以...自居
set out 出发, 装饰, 开始
set out for 出发去...
set the record straight 澄清问题
set up 竖立, 建立, 创立, 建造
set upon 攻击, 开始

2. 一套, 一副, 一部, 布景, 收音机, 电视机, 摄影场

中文（繁體） (Chinese (Traditional))
1.
n. - 一套, 一副, 一部, 佈景, 收音機, 電視機, 攝影場

2.
v. tr. - 放, 置, 使接觸, 豎立, 使處於
v. intr. - 落, 下沈, 凝結, 凝固, 衰落, 固定, 定型
adj. - 固定的, 堅決的, 規定的
n. - 一套, 一批, 一副

idioms:

be set in one's ways 固執
set about 著手, 開始
set against 反對, 使對立, 使平衡, 從...扣除, 比較
set back 使受挫折
set by 把...留開
set down 放下, 制定
set down as 認為..., 把...看作
set eyes on 看到, 望見
set great store by 很相信
set in 開始, 上漲, 到來
set in concrete 固定
set off 出發, 使爆炸, 動身
set on 攻擊, 慫恿, 前進
set one's house in order 整理
set oneself up as 以...自居
set out 出發, 裝飾, 開始
set out for 出發去...
set the record straight 澄清問題
set up 豎立, 建立, 創立, 建造
set upon 攻擊, 開始

한국어 (Korean)
1.
v. tr. - ~을 놓다, 가격을 매기다, (모종, 씨 등을) 심다
v. intr. - (해, 달 등이) 저물다, (일을) 시작하다, 굳어지다

idioms:

be set in one's ways 자기 방식 등에 집착하다
set about ~에 착수하다, ~을 공격하다, 퍼뜨리다
set against ~와 비교하다, ~와 사이가 틀어지게 하다
set back 저지하다, (시계 바늘 등을) 되돌리다, ~에 비용을 들이다
set by 제거하다, 저축하다, 몹시 귀하게 여기다
set down 밑에 놓다, 적어두다, 규정하다
set down as ~로 보다
set eyes on ~을 보다, 발견하다
set in 일어나다, 유행되다
set off 돋보이게 하다, 벌충하다, 구획하다
set on ~을 덮치다, (개 따위를) 공격 시키다, 부추기다
set oneself up as ~을 자처하다, ~이라고 주장하다
set out 출발하다, 말하다, 두드러지게 하다
set out for 출발하다
set upon ~에 덮쳐 들다

2.
n. - (해 등의) 짐, 한 벌, (라디오) 수신기

3.
adj. - 고정된, 결심한, 정해진

日本語 (Japanese)
v. - 置く, する, 課する, 示す, 付ける, 配置する, 決める, 調整する, 準備する, 沈む, 与える, 固める, 固まる, セットする, 向ける
n. - 一組, 仲間, 格好, 着心地, 受信機, 受像機, 回, 流れ, 傾向, 苗, 舞台装置
adj. - 定められた, 指定の, 型にはまった, 意志が固い, 頑固な, 固定した, 硬直した, 定食の

idioms:

set about …し始める, 攻撃する
set by 取り除く, しまっておく
set down 下に置く, 着陸させる, 書き留める, せいにする, みなす, 着陸する
set down as ～とみなす
set down on ～に降ろす
set in 始まる, 岸のほうへ吹く, 差してくる
set light/fire to 火をつける, 興奮させる
set off 出発する, 爆発させる, 上げる, 引き起こす, 始動させる, 引き立たせる, 始めさせる, …に…させ始める, 釣り合わせる
set on 攻撃する, 注ぐ, 前進する
set out 出発する, …するつもりで始める, はっきりと述べる, 広げる, 植え付ける, 設計する
set out for 出発する
set up 立てる, 組み立てる, 設立する, 主張する, 始めさせる, 開業させる, 供給する, 起こす

العربيه (Arabic)
‏(فعل) يطلق, ينصب, يهيأ, يقعد (الاسم) مجموعه من ألكتب أو ألأرقام تؤلف سلسله تامه (صفه) معين, محدد, متلاحم, عنيف, ضار‏

עברית (Hebrew)
v. tr. - ‮הניח, שם, הציב, קבע, עורר, גרם, ערך, סידר, הכין, הטיל על, כיוון, ויסת, המריץ‬
v. intr. - ‮הקריש, הפסיק לנוע, עטו ארשת קשה (פנים), חש או הראה נטייה מסוימת, נעשה לפרי (פרח), נעה בכיוון מסוים (גיאות), עשה פרי (עץ), ישב (עגה), התחיל‬
n. - ‮אנשים, חוג, קבוצה, מבנה, תנוחה, נטייה, התקרשות, מערכת כלי-בית, מקלט (מ פתוחה), שתיל, מערכת, מערכה (טניס), קבוצת עצמים בעלי תכונות משותפות או המשמשים למטרה אחת, קבוצת תלמידים בעלי יכולת ממוצעת דומה, נצר או פקעת לשתילה, תנוחה טבעית‬
adj. - ‮יציב, קבוע מראש, עקשני, נחוש-דיעה, מוכן, ערוך, מוכן לפעולה, נקבע מראש, מומלץ‬

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set: Definition, Synonyms and Much More from Answers.com

Describing sets

Membership

Cardinality

Subsets

Power set

Special sets

Basic operations

Unions

Intersections

Complements

Cartesian product

Applications

Axiomatic set theory

See also

Notes

References

External links