This article gives an introduction to what mathematicians call "intuitive" or "naive" set theory; for a more detailed account see Naive set theory. Naive set theory is one of several theories of sets used in the discussion of the Foundations of mathematics. For a rigorous modern axiomatic treatment of sets, see Axiomatic set theory. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject

A set is a collection of distinct objects considered as a whole. Sets are one of the most fundamental concepts in mathematics. The term "concept" is traced back to 1554–60 ( l conceptum - something conceived but what is today termed "the classical theory of concepts" is the theory of Aristotle Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and 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. The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar Mathematics education is a term that refers both to the practice of Teaching and Learning Mathematics, as well as to a field of scholarly Research See also Primary education A primary school (from French école primaire) is an institution where children receive the first stage of Compulsory 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 ^{[1][2] [3] [4]} 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. Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language For other uses see Abstract In Philosophy it is commonly considered that every object is either abstract or concrete The type versus token distinction separates an abstract concept from the objects which are particular instances of the concept The musical instrument is spelled Cymbal. A symbol is something --- such as an object, Picture, written word a sound a piece However, proponents of mathematical realism including Penelope Maddy have argued that sets are concrete objects. The philosophy of mathematics is the branch of Philosophy that studies the philosophical assumptions foundations and implications of Mathematics. Penelope Maddy is a UCI Distinguished Professor of Logic and Philosophy of Science and of Mathematics at the University of California Irvine In Physics, a physical body (sometimes called simply a body or even an object) is a collection of Masses taken to be one

Contents 1 Definition 2 Describing sets 3 Membership 4 Cardinality 5 Subsets 5.1 Power set 6 Special sets 7 Basic operations 7.1 Unions 7.2 Intersections 7.3 Complements 7.4 Cartesian product 8 Applications 9 Axiomatic set theory 10 See also 11 Notes 12 References 13 External links

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:^[5]

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. Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia.

The elements of a set, also called its members, can be anything: numbers, people, letters of the alphabet, other sets, and so on. In Mathematics, the elements or members of a set (or more generally a class) are all those objects which when collected together make up the Sets are conventionally denoted with capital letters. Capital letters or majuscules pronunciation /məˈdʒʌskyuls ˈmædʒəˌskyuls/ in the Roman alphabet A, B, C, D, 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. In Mathematics, a multiset (or bag) is a generalization of a set. 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. In Mathematics, a sequence is an ordered list of objects (or events In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the 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. In Logic and Mathematics, an intensional definition gives the meaning of a term by specifying all the properties required to come to that Definition Semantics is the study of meaning in communication The word derives from Greek σημαντικός ( semantikos) "significant" from See this example:

A is the set whose members are the first four positive integers. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

B is the set of colors of the French flag. The National flag of France (known in French as drapeau tricolore, drapeau français,and in military parlance les couleurs

The second way is by extension, that is, listing each member of the set. In any of several studies that treat the use of signs for example in Linguistics, Logic, Mathematics, Semantics, and Semiotics, the 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. An extensional definition of a concept or term formulates its meaning by specifying its extension, that is every object that falls under the Definition Brackets are Punctuation marks used in pairs to set apart or interject text within other text 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 (". Ellipsis (plural ellipses; from Greek 'omission' in Printing and Writing refers to a mark or series of marks that usually indicate an intentional . . ") 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, . In Mathematics, the parity of an object states whether it is even or odd . . }.

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. In Set theory and its applications to Logic, Mathematics, and Computer science, set-builder notation (sometimes simply "set notation" This article refers to the REM live recording For the mathematical term see Perfect square. " Sometimes the vertical bar ("|") is used instead of the colon. Note "broken bar" and the glyph "¦" redirect here

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

Membership

Main article: Element (mathematics)

If something is or is not an element of a particular set then this is symbolised by ∈ and ∉ respectively. In Mathematics, the elements or members of a set (or more generally a class) are all those objects which when collected together make up the So, with respect to the sets defined above:

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

Cardinality

Main article: Cardinality

The cardinality |S| of a set S is "the number of members of S. In Mathematics, the cardinality of a set is a measure of the "number of elements of the set" " 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. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members 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. In Set theory, an infinite set is a set that is not a Finite set. The set N of natural numbers, for instance, is infinite. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an Some infinite cardinalities are greater than others. For instance, the set of real numbers has greater cardinality than the set of natural numbers. In Mathematics, the real numbers may be described informally in several different ways 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. In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points

Subsets

Main article: Subset

If every member of set A is also a member of set B, then A is said to be a subset of B, written (also pronounced A is contained in B). Equivalently, we can write , read as B is a superset of A, B includes A, or B contains A. The relationship between sets established by is called inclusion or containment. This article sets out the set-theoretic notion of relation For a more elementary point of view see Binary relations and Triadic relations

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

Note that the expressions and are used differently by different authors; some authors use them to mean the same as (respectively ), whereas other use them to mean the same as (respectively ).

Example:

• The set of all men is a proper subset of the set of all people.
• 
• 

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

• 
• 

Power set

Main article: Power set

The power set of a set S can be defined as the set of all subsets of S. In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( 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 the cardinality of the power set is 2³, or 8. 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. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects

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. Blackboard bold is a Typeface style often used for certain symbols in Mathematics and Physics texts in which certain lines of the symbol (usually vertical Special sets of numbers include:

• , denoting the set of all primes. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1
• , denoting the set of all natural numbers. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an That is to say, = {1, 2, 3, . . . }.
• , denoting the set of all whole numbers. That is to say, = {0, 1, 2, 3, . . . }.
• , denoting the set of all integers (whether positive, negative or zero). The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French So = {. . . , -2, -1, 0, 1, 2, . . . }.
• , denoting the set of all rational numbers (that is, the set of all proper and improper fractions). In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object So, . For example, and . All integers are in this set since every integer a can be expressed as the fraction .
• , denoting the set of all real numbers. In Mathematics, the real numbers may be described informally in several different ways 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). In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction
• , denoting the set of all complex numbers. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

Each of these sets of numbers has an infinite number of elements, and . The primes are used less frequently than the others outside of number theory and related fields. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes

Basic operations

Unions

Main article: Union (set theory)

There are ways to construct new sets from existing ones. In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets Two sets can be "added" together. The union of A and B, denoted by A ∪ B, is the set of all things which are members of either A or B.

Examples:

• {1, 2} ∪ {red, white} = {1, 2, red, white}
• {1, 2, green} ∪ {red, white, green} = {1, 2, red, white, green}
• {1, 2} ∪ {1, 2} = {1, 2}.

Some basic properties of unions are:

• A ∪ B   =   B ∪ A
• A  ⊆  (A ∪ B)
• A ∪ A   =  A
• A ∪ ø   =  A
• A  ⊆  B if and only if A ∪ B = B. ↔

Intersections

Main article: Intersection (set theory)

A new set can also be constructed by determining which members two sets have "in common". In Mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently 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.

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

Main article: Complement (set theory)

Two sets can also be "subtracted". In Discrete mathematics and predominantly in Set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation 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 Mathematical logic, the universe of a structure (or model) is its domain. In such cases, U \ A, is called the absolute complement or simply complement of A, and is denoted by A′.

Examples:

• {1, 2} \ {red, white} = {1, 2}
• {1, 2, green} \ {red, white, green} = {1, 2}
• {1, 2} \ {1, 2} = ∅
• 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 ∪ A′ = U
• A ∩ A′ = ∅
• (A′ )′ = A
• A \ A = ∅
• A \ B = A ∩ B′.

Cartesian product

Main article: Cartesian product

A new set can be constructed by associating every element of one set with every element of another set. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. 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. In Mathematics, an ordered pair is a collection of two distinguishable objects one of which is identified as the first coordinate (or the first entry

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 × ∅ = ∅
• A × (B ∪ C) = (A × B) ∪ (A × C)
• |A × B| = |A| × |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. In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real

One of the main applications of naive set theory is constructing relations. This article sets out the set-theoretic notion of relation For a more elementary point of view see Binary relations and Triadic relations A relation from a domain A to a codomain B is nothing but a subset of A × B. In Mathematics, the codomain, or target, of a function f: X → Y is the set 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 and a codomain set that is also , 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, its corresponding ordered pair, (y, y²) is a member of the set F.

Axiomatic set theory

Main article: 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. Naive set theory is one of several theories of sets used in the discussion of the Foundations of mathematics. 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. Part of the Foundations of mathematics, Russell's paradox (also known as Russell's antinomy) discovered by Bertrand Russell in 1901 showed that the e. the "set" does not exist.
• Cantor's paradox - It shows that "the set of all sets" cannot exist. In Set theory, Cantor's paradox is the Theorem that there is no greatest Cardinal number, so that the collection of "infinite sizes" is itself

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. First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science

For most purposes however, the naive set theory is still useful. Naive set theory is one of several theories of sets used in the discussion of the Foundations of mathematics.

See also

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. Harvard University Press ( HUP) is a Publishing house, a division of Harvard University, that is highly respected in Academic publishing.
• Halmos, Paul R., Naive Set Theory, Princeton, N. Paul Richard Halmos ( March 3 1916 &mdash October 2 2006) was a Hungarian -born Jewish American Mathematician 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. Dover Publications is an American book Publisher founded in 1941 by Hayward Cirker and his wife Blanche

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

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