• Danish: størkne
• Dutch: harden
• Finnish: jähmettyä, kovettua, asettua
• French: geler
• German: härten, aushärten, fest werden
• Interlingua: solidificar se
• Portuguese: solidificar
• Swedish: stelna
• Vietnamese: đặt lại, se lại

• Czech: zapadnout
• Danish: gå ned
• Dutch: ondergaan
• Esperanto: subiri
• Finnish: laskea
• French: coucher
• German: untergehen
• Hebrew: לשקוע
• Slovene: zaiti
• Spanish: poner
• Swedish: gå ned, gå ner
• Vietnamese: lặn

• ttbc Bulgarian: поставям (postavjam), слагам (slagam) (1); втвърдявам (vtvǎrdjavam) (1), започвам (započvam), почвам (počvam) (2)
• ttbc Hungarian: tenni (1)
• ttbc Indonesian: mengatur, memasang, mempaskan, menepatkan
• ttbc Interlingua: poner, deponer (1), clavar (2), arrangiar, preparar (3)
• ttbc Portuguese: pôr, colocar (1); cravar, pregar (2); pôr, arrumar, preparar (3)
• ttbc Romanian: seta, instala
• ttbc Slovak: položiť (1); usadnúť (1), zapadnúť (3)
• ttbc Korean: 배치하다
• ttbc Spanish: cuajar, fraguar

Noun

1. A matching collection of similar things.

   a set of tables

2. A collection of various objects for a particular purpose.

   a set of tools

3. An object made up several parts

   a set of steps

4. A well-defined collection of mathematical objects, its elements, often having a common property.
5. In the context of "in plural, “sets”|mathematics}}

this mathematical sets

This article gives an 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.

A set is a collection of distinct objects considered as a whole. Sets are one of the most fundamental concepts in mathematics. The study of the structure of 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 a foundation from which nearly all of mathematics can be derived.

In philosophy, sets are ordinarily considered to be abstract objects

the physical tokens of which are, for instance; three cups on a table when spoken of together as "the cups", or the chalk lines on a board in the form of the opening and closing curly bracket symbols along with any other symbols in between the two bracket symbols. However, proponents of mathematical realism including Penelope Maddy have argued that sets are concrete objects.

Definition

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: 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 =

D =

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,

= =

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:

,

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 .

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 = 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 =

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 ∈ and ∉ respectively. So, with respect to the sets defined above:

• 4 ∈ A and 285 ∈ F (since 285 = 17² − 4); but
• 9 ∉ F and green ∉ B.

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).

• Axiomatic set theory
• Class (set theory)
• Dense set
• Family (mathematics)
• Fuzzy set
• Mathematical structure
• Multiset
• Naive set theory
• Rough set
• Russell's paradox
• Scientific classification
• Taxonomy
• Tuple

Notes

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

• C2 Wiki - Examples of set operations using English operators. Dead Link
• Set of Sets articles on Plainmath.net- A series of articles on all aspects of sets

portalpar Logic

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