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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 important and fundamental concepts in modern mathematics. The study of the structure of possible sets, set theory, is rich and ongoing. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... For other uses, see Concept (disambiguation). ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...

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. Alternative meaning: Nineteenth Century (periodical) (18th century &#8212; 19th century &#8212; 20th century &#8212; more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... 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 on this practice. ... Primary or elementary education is the first years of formal, structured education that occurs during childhood. ...

This article gives a brief and 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. Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Intuition is an unconscious form of knowledge. ... The word theory has a number of distinct meanings in different fields of knowledge, depending on their methodologies and the context of discussion. ... In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ... An axiom is a sentence or proposition that is not proved or demonstrated and is considered as obvious or as an initial necessary consensus for a theory building or acceptation. ... This article or section is in need of attention from an expert on the subject. ...

The intersection of two sets is made up of the objects contained in both sets.

Image File history File links Venn_A_intersect_B.svgâ€Ž  Summary  Licensing File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Mathematics ... Image File history File links Venn_A_intersect_B.svgâ€Ž  Summary  Licensing File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Mathematics ...

At the beginning of his work Beiträge zur Begründung der transfiniten Mengenlehre, Georg Cantor, the principal creator of set theory, gave the following definition of a set:[1] Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. ...

 “ By a set we understand any collection M of definite, distinct objects m of our perception or of our thought (which will be called the elements of M) into a whole. ”

The objects of a set are also called its members. The elements of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with capital letters, for instance A, B and C. Two sets A and B are said to be equal if every member of A is also a member of B and crucially, every member of B is also a member of A; this is written A = B. In mathematics, an element (also called a member) is an object contained in a set (or more generally a class). ... Capital letters or majuscules (in the Roman alphabet: A, B, C, ...) are one type of case in a writing system. ...

A set, unlike a multiset, cannot contain two or more identical elements. All set operations preserve the property that each element in the set is unique. Similarly, the order in which the elements of a set are listed is irrelevant, unlike a sequence or tuple. In mathematics, a multiset (sometimes also called a bag) differs from a set in that each member has a multiplicity, which is a natural number indicating (loosely speaking) how many times it is a member, or perhaps how many memberships it has in the multiset. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In mathematics, a tuple is a finite sequence (also known as an ordered list) of objects, each of a specified type. ...

## Describing sets

Main article: Set-builder notation

Not all sets have precise descriptions; they may be arbitrary collections, with no expressible inclusion criteria. In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by indicating the properties that its members must satisfy. ...

Some sets may be described in words:

A is the set whose members are the first four positive whole numbers.
B is the set whose members are the colors of the French flag.

By convention, a set can be defined by explicitly listing its elements between braces (sometimes called curly brackets or curly braces): The whole numbers are the nonnegative integers (0, 1, 2, 3, ...) The set of all whole numbers is represented by the symbol = {0, 1, 2, 3, ...} Algebraically, the elements of form a commutative monoid under addition (with identity element zero), and under multiplication (with identity element one). ... Flag Ratio: 2:3 The national flag of France (Vexillological symbol: , known in French as drapeau tricolore, drapeau bleu-blanc-rouge, drapeau franÃ§ais, rarely, le tricolore and, in military parlance, les couleurs) is a tricolour featuring three vertical bands coloured blue (hoist side), white, and red. ... For technical reasons, :) and some similar combinations starting with : redirect here. ...

C = {4, 2, 1, 3}
D = {red, white, blue}

Two different descriptions may define the same set. Using the above examples, A and C are identical, since they have precisely the same members. The shorthand A = C is used to express this equality. In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ...

The set described by set builder notation does not depend on the order in which the elements are listed, nor on whether there are repetitions in the list. For example,

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

This is because the notation { ... } merely indicates that the set being described includes each element listed; if an element is listed more than once, or if two elements are transposed, this has no effect on the resulting set.

For sets with many elements, an abbreviated list can sometimes be used. The first one thousand positive whole numbers can be described using the symbolic shorthand:

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

where the ellipsis (...) indicates the list continues in the same way. Ellipses may also be used where sets extend to infinity; the set of positive even numbers can be described: {2, 4, 6, 8, ... }. Distinguish from ellipse. ... In mathematics, any integer (whole number) is either even or odd. ...

Sets, particularly more complex ones, can use a different notation. The set F, whose members are the first twenty numbers which are four less than a square integer, can be described:

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

In this description, the colon (:) means such that, and the description can be interpreted as "F is the set of numbers of the form n2 – 4, such that n is a whole number in the range from 0 to 19 inclusive." Sometimes the pipe (|) is used instead of the colon. The integers are commonly denoted by the above symbol. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ...

## Membership

Main article: Element (mathematics)

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: In mathematics, an element (also called a member) is an object contained in a set (or more generally a class). ...

• $4 in A$ and $285 in F$ (since 285 = 17² − 4); but
• $9 notin F$ and $mathrm{green} notin B$.

## Cardinality

Main article: Cardinality

In mathematics, the cardinality of a set is a measure of the "number of elements of the set". Each of the sets described above has a definite number of members; for example, the set A has four members, while the set B has three members, denoted |A|=4 and |B|=3 respectively. In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality â€“ one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ...

A set can have zero members. Such a set is called the empty set (or the null set) and is denoted by the symbol ø. Letting A be the set of all three-sided squares, it has zero members, and thus A = ø. Like the number zero, though seemingly trivial, the empty set turns out to be quite important in mathematics. The empty set is the set containing no elements. ... 0 (zero) is both a number and a numerical digit used to represent that number in numerals. ...

A set can have an infinite number of members; for example, the set of natural numbers is infinite. Some infinite sets have larger cardinality than others; for instance, real numbers have larger cardinality than natural numbers. However, counter-intuitively, it can be shown that the number of points in a straight line is the same as the number of points in a segment of that line, or in a plane, or in a n-dimensional Euclidean space. The infinity symbol âˆž in several typefaces. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ... Look up plane in Wiktionary, the free dictionary. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

## 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 $A subseteq B$ (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. A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In mathematics, the concept of a relation is a generalization of 2-place relations, such as the relation of equality, denoted by the sign = in a statement like 5 + 7 = 12, or the relation of order, denoted by the sign < in a statement like 5 < 12. Relations that involve two...

If A is a subset of, but not equal to, B, then A is called a proper subset of B, written $A subset B$ (A is a proper subset of B) or $B supset A$ (B is proper superset of A). However, in some literature these symbols are read the same as $subseteq$ and $supseteq$, so the more explicit symbols $subsetneq$ and $supsetneq$ are often used for proper subsets and supersets.

A is a subset of B

Example: Image File history File links Venn_A_subset_B.svgâ€Ž Venn diagram for A is a subset of B. Modification of Image:Venn A intersect B.svg based on w:en:Image:Venn A subset B.png File links The following pages on the English Wikipedia link to this file (pages on other...

• The set of all men is a proper subset of the set of all people.
• ${1,3} subset {1,2,3,4}$
• ${1, 2, 3, 4} subseteq {1,2,3,4}$

The empty set is a subset of every set and every set is a subset of itself: A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ...

• $emptyset subseteq A$
• $A subseteq A$

### Power set

Main article: 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 2S. In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... In mathematics the term countable set is used to describe the size of a set, e. ... In mathematics, an uncountable set is a set which is not countable. ...

## 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: An example of blackboard bold letters. ...

• $mathbb{P}$, 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, $mathbb{Q} = left{ begin{matrix}frac{a}{b} end{matrix}: a,b in mathbb{Z}, b neq 0right}$. 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. In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... The integers are commonly denoted by the above symbol. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ... In algebra, an improper fraction is a fraction where the absolute value of the numerator is greater than the absolute value of the denominator. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, an irrational number is any real number that is not a rational number â€” that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... 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 of problems that arise from their study. ...

## Basic operations

### Unions

Main article: Union (set theory)

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. In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...

The union of A and B

Examples: Venn diagram for A union B. Created by me: Paul August 02:48, Aug 24, 2004 (UTC) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

• {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

### 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. 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, all elements of B that also belong to A), but no other elements. ...

The intersection of A and B

Examples: Image File history File links Venn_A_intersect_B.svgâ€Ž  Summary  Licensing File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Mathematics ...

• {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

### 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 set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...

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′. In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use. ...

The relative complement
of A in B
The complement of A in U

Examples: Venn diagram for the relative complement of A in B. Also called the set theoretic difference of B and A, denoted by B &#8722; A. Image created and uploaded by: Paul August 12:31, Aug 25, 2004 (UTC). ... Venn diagram for the set-theoretic complement of A in the universal set U, denoted by A&#8242;. Image created and uploaded by: Paul August 22:33, Aug 24, 2004 (UTC) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old...

• {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′

## 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 universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... This picture illustrates how the hours on a clock form a group under modular addition. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

## 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. It was found that this definition spawned several paradoxes, most notably: This article or section is in need of attention from an expert on the subject. ... In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...

• Russell's paradox - It shows that the "set of all sets which do not contain themselves," i.e. the "set" $left { x: xmbox{ is a set and }xnotin x right }$ 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 since entire 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. Part of the foundation of mathematics, Russells paradox (also known as Russells antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction. ... Cantors paradox, also known as the paradox of the greatest cardinal, demonstrates that there is no cardinal greater than all other cardinalsâ€”that the class of cardinal numbers is infinite. ... First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ... This article or section is in need of attention from an expert on the subject. ...

For most purposes however, the naive set theory is still useful. In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...

An alternative set theory is an alternative mathematical approach to the concept of set. ... This article or section is in need of attention from an expert on the subject. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if, intuitively, any point in X can be well-approximated by points in A. Formally, A is dense in X if for any point x in X, any neighborhood of... In mathematics, an index set is another name for a function domain. ... Fuzzy sets are an extension of classical set theory and are used in fuzzy logic. ... In mathematics, a structure on a set is some additional mathematical objects that, loosely speaking, attach to the set, making it easier to visualize or work with. ... In mathematics, a multiset (sometimes also called a bag) differs from a set in that each member has a multiplicity, which is a natural number indicating (loosely speaking) how many times it is a member, or perhaps how many memberships it has in the multiset. ... Cover of Rough Sets: Theoretical Aspects of Reasoning about Data by ZdzisÅ‚aw Pawlak (Kluwer 1991). ... Part of the foundation of mathematics, Russells paradox (also known as Russells antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction. ... Scientific classification or biological classification is a method by which biologists group and categorize species of organisms. ... Look up taxonomy in Wiktionary, the free dictionary. ... In mathematics, a tuple is a finite sequence (also known as an ordered list) of objects, each of a specified type. ...

## Notes and references

1. ^ Allenby, p. 1.

• 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
• Allenby, R.B.J.T, Rings, Fields and Groups, Leeds, England: Butterworth Heinemann (1991) ISBN 0-340-54440-6

Dover Publications is a book publisher founded in 1941. ...

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