TheInfoList

Naive set theory is any of several theories of sets used in the discussion of the
foundations of mathematics Foundations of mathematics is the study of the philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical or mental r ...
. Unlike axiomatic set theories, which are defined using
formal logic Logic is an interdisciplinary field which studies truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to re ...
, naive set theory is defined informally, in
natural language In neuropsychology Neuropsychology is a branch of psychology. It is concerned with how a person's cognition and behavior are related to the brain and the rest of the nervous system. Professionals in this branch of psychology often focus on ...
. It describes the aspects of mathematical sets familiar in
discrete mathematics Discrete mathematics is the study of mathematical structures In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geom ...
(for example
Venn diagram A Venn diagram is a widely used diagram A diagram is a symbolic representation Representation may refer to: Law and politics *Representation (politics) Political representation is the activity of making citizens "present" in public policy ...
s and symbolic reasoning about their
Boolean algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
), and suffices for the everyday use of set theory concepts in contemporary mathematics. Sets are of great importance in mathematics; in modern formal treatments, most mathematical objects (
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

s,
relations Relation or relations may refer to: General uses *International relations, the study of interconnection of politics, economics, and law on a global level *Interpersonal relationship, association or acquaintance between two or more people *Public ...
,
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
, etc.) are defined in terms of sets. Naive set theory suffices for many purposes, while also serving as a stepping-stone towards more formal treatments.

# Method

A ''naive theory'' in the sense of "naive set theory" is a non-formalized theory, that is, a theory that uses
natural language In neuropsychology Neuropsychology is a branch of psychology. It is concerned with how a person's cognition and behavior are related to the brain and the rest of the nervous system. Professionals in this branch of psychology often focus on ...
to describe sets and operations on sets. The words ''and'', ''or'', ''if ... then'', ''not'', ''for some'', ''for every'' are treated as in ordinary mathematics. As a matter of convenience, use of naive set theory and its formalism prevails even in higher mathematics – including in more formal settings of set theory itself. The first development of
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
was a naive set theory. It was created at the end of the 19th century by
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...
as part of his study of
infinite set In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...
s and developed by
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He worked as a mathematics professor at the University of Jena, and is understood by many to be the father of analy ...
in his ''Grundgesetze der Arithmetik''. Naive set theory may refer to several very distinct notions. It may refer to * Informal presentation of an axiomatic set theory, e.g. as in ''
Naive Set Theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sens ...
'' by
Paul Halmos Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a HungarianHungarian may refer to: * Hungary, a country in Central Europe * Kingdom of Hungary, state of Hungary, existing between 1000 and 1946 * Hungarians, ethnic g ...
. * Early or later versions of
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...
's theory and other informal systems. * Decidedly inconsistent theories (whether axiomatic or not), such as a theory of
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He worked as a mathematics professor at the University of Jena, and is understood by many to be the father of analy ...
that yielded
Russell's paradox In the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theori ...

, and theories of
Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

and
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory In algebra, ring theory is the study of ring (mathematics), rings ...
.

The assumption that any property may be used to form a set, without restriction, leads to
paradox A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...

es. One common example is
Russell's paradox In the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theori ...

: there is no set consisting of "all sets that do not contain themselves". Thus consistent systems of naive set theory must include some limitations on the principles which can be used to form sets.

## Cantor's theory

Some believe that
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...
's set theory was not actually implicated in the set-theoretic paradoxes (see Frápolli 1991). One difficulty in determining this with certainty is that Cantor did not provide an axiomatization of his system. By 1899, Cantor was aware of some of the paradoxes following from unrestricted interpretation of his theory, for instance
Cantor's paradox In set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ar ...
Letter from Cantor to
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, G ...
on September 26, 1897, p. 388.
and the
Burali-Forti paradox In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ty ...
,Letter from Cantor to
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory In algebra, ring theory is the study of ring (mathematics), rings ...
on August 3, 1899, p. 408.
and did not believe that they discredited his theory.Letters from Cantor to
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory In algebra, ring theory is the study of ring (mathematics), rings ...
on August 3, 1899 and on August 30, 1899, p. 448 (System aller denkbaren Klassen) and p. 407. (There is no set of all sets.)
Cantor's paradox can actually be derived from the above (false) assumption—that any property may be used to form a set—using for " is a
cardinal number 150px, Aleph null, the smallest infinite cardinal In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...
". Frege explicitly axiomatized a theory in which a formalized version of naive set theory can be interpreted, and it is ''this'' formal theory which
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose know ...

## Axiomatic theories

Axiomatic set theory was developed in response to these early attempts to understand sets, with the goal of determining precisely what operations were allowed and when.

## Consistency

A naive set theory is not ''necessarily'' inconsistent, if it correctly specifies the sets allowed to be considered. This can be done by the means of definitions, which are implicit axioms. It is possible to state all the axioms explicitly, as in the case of Halmos' ''Naive Set Theory'', which is actually an informal presentation of the usual axiomatic
Zermelo–Fraenkel set theory In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any t ...
. It is "naive" in that the language and notations are those of ordinary informal mathematics, and in that it does not deal with consistency or completeness of the axiom system. Likewise, an axiomatic set theory is not necessarily consistent: not necessarily free of paradoxes. It follows from
Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (nu ...
that a sufficiently complicated
first order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal system A formal system is used for inferring theorems from axioms according to a set of rules. These rules, wh ...
system (which includes most common axiomatic set theories) cannot be proved consistent from within the theory itself – even if it actually is consistent. However, the common axiomatic systems are generally believed to be consistent; by their axioms they do exclude ''some'' paradoxes, like
Russell's paradox In the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theori ...

. Based on Gödel's theorem, it is just not known – and never can be – if there are ''no'' paradoxes at all in these theories or in any first-order set theory. The term ''naive set theory'' is still today also used in some literature to refer to the set theories studied by Frege and Cantor, rather than to the informal counterparts of modern axiomatic set theory.

## Utility

The choice between an axiomatic approach and other approaches is largely a matter of convenience. In everyday mathematics the best choice may be informal use of axiomatic set theory. References to particular axioms typically then occur only when demanded by tradition, e.g. the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

is often mentioned when used. Likewise, formal proofs occur only when warranted by exceptional circumstances. This informal usage of axiomatic set theory can have (depending on notation) precisely the ''appearance'' of naive set theory as outlined below. It is considerably easier to read and write (in the formulation of most statements, proofs, and lines of discussion) and is less error-prone than a strictly formal approach.

# Sets, membership and equality

In naive set theory, a set is described as a well-defined collection of objects. These objects are called the elements or members of the set. Objects can be anything: numbers, people, other sets, etc. For instance, 4 is a member of the set of all even
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s. Clearly, the set of even numbers is infinitely large; there is no requirement that a set be finite. The definition of sets goes back to
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...
. He wrote in his 1915 article
Beiträge zur Begründung der transfiniten Mengenlehre
':
“Unter einer 'Menge' verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten unserer Anschauung oder unseres Denkens (welche die 'Elemente' von M genannt werden) zu einem Ganzen.” – Georg Cantor
“A set is a gathering together into a whole of definite, distinct objects of our perception or of our thought—which are called elements of the set.” – Georg Cantor

## Note on consistency

It does ''not'' follow from this definition ''how'' sets can be formed, and what operations on sets again will produce a set. The term "well-defined" in "well-defined collection of objects" cannot, by itself, guarantee the consistency and unambiguity of what exactly constitutes and what does not constitute a set. Attempting to achieve this would be the realm of axiomatic set theory or of axiomatic class theory. The problem, in this context, with informally formulated set theories, not derived from (and implying) any particular axiomatic theory, is that there may be several widely differing formalized versions, that have both different sets and different rules for how new sets may be formed, that all conform to the original informal definition. For example, Cantor's verbatim definition allows for considerable freedom in what constitutes a set. On the other hand, it is unlikely that Cantor was particularly interested in sets containing cats and dogs, but rather only in sets containing purely mathematical objects. An example of such a class of sets could be the
von Neumann universe In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ty ...
. But even when fixing the class of sets under consideration, it is not always clear which rules for set formation are allowed without introducing paradoxes. For the purpose of fixing the discussion below, the term "well-defined" should instead be interpreted as an ''intention'', with either implicit or explicit rules (axioms or definitions), to rule out inconsistencies. The purpose is to keep the often deep and difficult issues of consistency away from the, usually simpler, context at hand. An explicit ruling out of ''all'' conceivable inconsistencies (paradoxes) cannot be achieved for an axiomatic set theory anyway, due to Gödel's second incompleteness theorem, so this does not at all hamper the utility of naive set theory as compared to axiomatic set theory in the simple contexts considered below. It merely simplifies the discussion. Consistency is henceforth taken for granted unless explicitly mentioned.

## Membership

If ''x'' is a member of a set ''A'', then it is also said that ''x'' belongs to ''A'', or that ''x'' is in ''A''. This is denoted by ''x'' ∈ ''A''. The symbol ∈ is a derivation from the lowercase Greek letter
epsilon Epsilon (, ; uppercase Letter case (or just case) is the distinction between the letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (or more formally ''minuscule'') in the written represe ...

, "ε", introduced by
Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

in 1889 and is the first letter of the wor
ἐστί
(means "is"). The symbol ∉ is often used to write ''x'' ∉ ''A'', meaning "x is not in A".

## Equality

Two sets ''A'' and ''B'' are defined to be
equal Equal or equals may refer to: Arts and entertainment * Equals (film), ''Equals'' (film), a 2015 American science fiction film * Equals (game), ''Equals'' (game), a board game * The Equals, a British pop group formed in 1965 * "Equal", a 2016 song b ...
when they have precisely the same elements, that is, if every element of ''A'' is an element of ''B'' and every element of ''B'' is an element of ''A''. (See
axiom of extensionality In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory. Formal statement In the formal language ...
.) Thus a set is completely determined by its elements; the description is immaterial. For example, the set with elements 2, 3, and 5 is equal to the set of all
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s less than 6. If the sets ''A'' and ''B'' are equal, this is denoted symbolically as ''A'' = ''B'' (as usual).

## Empty set

The
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

, often denoted Ø and sometimes $\$, is a set with no members at all. Because a set is determined completely by its elements, there can be only one empty set. (See
axiom of empty set In axiomatic set theory illustrating the intersection of two sets. Set theory is a branch of mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) include ...
.) Although the empty set has no members, it can be a member of other sets. Thus Ø ≠ , because the former has no members and the latter has one member. In mathematics, the only sets with which one needs to be concerned can be built up from the empty set alone.

# Specifying sets

The simplest way to describe a set is to list its elements between curly braces (known as defining a set ''extensionally''). Thus denotes the set whose only elements are and . (See
axiom of pairing In axiomatic set theory illustrating the intersection of two sets. Set theory is a branch of mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includ ...
.) Note the following points: *The order of elements is immaterial; for example, . *Repetition ( multiplicity) of elements is irrelevant; for example, . (These are consequences of the definition of equality in the previous section.) This notation can be informally abused by saying something like to indicate the set of all dogs, but this example would usually be read by mathematicians as "the set containing the single element ''dogs''". An extreme (but correct) example of this notation is , which denotes the empty set. The notation , or sometimes , is used to denote the set containing all objects for which the condition holds (known as defining a set ''intensionally''). For example, denotes the set of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, denotes the set of everything with blonde hair. This notation is called
set-builder notation In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...
(or "set comprehension", particularly in the context of
Functional programming In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , ...
). Some variants of set builder notation are: * denotes the set of all that are already members of such that the condition holds for . For example, if is the set of
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s, then is the set of all even integers. (See
axiom of specification In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any ...
.) * denotes the set of all objects obtained by putting members of the set into the formula . For example, is again the set of all even integers. (See
axiom of replacement In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any t ...
.) * is the most general form of set builder notation. For example, is the set of all dog owners.

# Subsets

Given two sets ''A'' and ''B'', ''A'' is a
subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of ''B'' if every element of ''A'' is also an element of ''B''. In particular, each set ''B'' is a subset of itself; a subset of ''B'' that is not equal to ''B'' is called a proper subset. If ''A'' is a subset of ''B'', then one can also say that ''B'' is a superset of ''A'', that ''A'' is contained in ''B'', or that ''B'' contains ''A''. In symbols, ''A'' ⊆ ''B'' means that ''A'' is a subset of ''B'', and ''B'' ⊇ ''A'' means that ''B'' is a superset of ''A''. Some authors use the symbols ⊂ and ⊃ for subsets, and others use these symbols only for ''proper'' subsets. For clarity, one can explicitly use the symbols ⊊ and ⊋ to indicate non-equality. As an illustration, let R be the set of real numbers, let Z be the set of integers, let ''O'' be the set of odd integers, and let ''P'' be the set of current or former
U.S. Presidents The president of the United States is the head of state and head of government of the United States, indirectly elected to a four-year Term of office, term by the American people through the United States Electoral College, Electoral College. ...

. Then ''O'' is a subset of Z, Z is a subset of R, and (hence) ''O'' is a subset of R, where in all cases ''subset'' may even be read as ''proper subset''. Not all sets are comparable in this way. For example, it is not the case either that R is a subset of ''P'' nor that ''P'' is a subset of R. It follows immediately from the definition of equality of sets above that, given two sets ''A'' and ''B'', ''A'' = ''B'' if and only if ''A'' ⊆ ''B'' and ''B'' ⊆ ''A''. In fact this is often given as the definition of equality. Usually when trying to
prove Proof may refer to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Formal sciences * Formal proof, a construct in proof theory * Mathematical proof, a co ...
that two sets are equal, one aims to show these two inclusions. The
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

is a subset of every set (the statement that all elements of the empty set are also members of any set ''A'' is
vacuously trueIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
). The set of all subsets of a given set ''A'' is called the
power set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of ''A'' and is denoted by $2^A$ or $P\left(A\right)$; the "''P''" is sometimes in a
script Script may refer to: Writing systems * Script, a distinctive writing system, based on a repertoire of specific elements or symbols, or that repertoire * Script (styles of handwriting) * Script (Unicode), historical and modern scripts as organise ...
font. If the set ''A'' has ''n'' elements, then $P\left(A\right)$ will have $2^n$ elements.

# Universal sets and absolute complements

In certain contexts, one may consider all sets under consideration as being subsets of some given
universal set In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ty ...
. For instance, when investigating properties of the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s R (and subsets of R), R may be taken as the universal set. A true universal set is not included in standard set theory (see
Paradoxes A paradox, also known as an antinomy, is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-con ...
below), but is included in some non-standard set theories. Given a universal set U and a subset ''A'' of U, the
complement A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be: * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase addi ...
of ''A'' (in U) is defined as :''A''C := . In other words, ''A''C ("''A-complement''"; sometimes simply ''A, "''A-prime''" ) is the set of all members of U which are not members of ''A''. Thus with R, Z and ''O'' defined as in the section on subsets, if Z is the universal set, then ''OC'' is the set of even integers, while if R is the universal set, then ''OC'' is the set of all real numbers that are either even integers or not integers at all.

# Unions, intersections, and relative complements

Given two sets ''A'' and ''B'', their union is the set consisting of all objects which are elements of ''A'' or of ''B'' or of both (see
axiom of union In axiomatic set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as ...
). It is denoted by ''A'' ∪ ''B''. The
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
of ''A'' and ''B'' is the set of all objects which are both in ''A'' and in ''B''. It is denoted by ''A'' ∩ ''B''. Finally, the
relative complement In set theory, the complement of a Set (mathematics), set , often denoted by (or ), are the Element (mathematics), elements not in . When all sets under consideration are considered to be subsets of a given set , the absolute complement of is t ...
of ''B'' relative to ''A'', also known as the set theoretic difference of ''A'' and ''B'', is the set of all objects that belong to ''A'' but ''not'' to ''B''. It is written as ''A'' \ ''B'' or ''A'' − ''B''. Symbolically, these are respectively :''A'' ∪ B := ; :''A'' ∩ ''B'' :=  =  = ; :''A'' \ ''B'' :=  = . The set ''B'' doesn't have to be a subset of ''A'' for ''A'' \ ''B'' to make sense; this is the difference between the relative complement and the absolute complement (''A''C = ''U'' \ ''A'') from the previous section. To illustrate these ideas, let ''A'' be the set of left-handed people, and let ''B'' be the set of people with blond hair. Then ''A'' ∩ ''B'' is the set of all left-handed blond-haired people, while ''A'' ∪ ''B'' is the set of all people who are left-handed or blond-haired or both. ''A'' \ ''B'', on the other hand, is the set of all people that are left-handed but not blond-haired, while ''B'' \ ''A'' is the set of all people who have blond hair but aren't left-handed. Now let ''E'' be the set of all human beings, and let ''F'' be the set of all living things over 1000 years old. What is ''E'' ∩ ''F'' in this case? No living human being is over 1000 years old, so ''E'' ∩ ''F'' must be the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

. For any set ''A'', the power set $P\left(A\right)$ is a
Boolean algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
under the operations of union and intersection.

# Ordered pairs and Cartesian products

Intuitively, an
ordered pair In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

is simply a collection of two objects such that one can be distinguished as the ''first element'' and the other as the ''second element'', and having the fundamental property that, two ordered pairs are equal if and only if their ''first elements'' are equal and their ''second elements'' are equal. Formally, an ordered pair with first coordinate ''a'', and second coordinate ''b'', usually denoted by (''a'', ''b''), can be defined as the set . It follows that, two ordered pairs (''a'',''b'') and (''c'',''d'') are equal if and only if ''a'' = ''c'' and ''b'' = ''d''. Alternatively, an ordered pair can be formally thought of as a set with a
total order In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. (The notation (''a'', ''b'') is also used to denote an
open interval In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
on the
real number line Real may refer to: Currencies * Brazilian real The Brazilian real ( pt, real, plural, pl. '; currency symbol, sign: R\$; ISO 4217, code: BRL) is the official currency of Brazil. It is subdivided into 100 centavos. The Central Bank of Brazil ...

, but the context should make it clear which meaning is intended. Otherwise, the notation ]''a'', ''b'' may be used to denote the open interval whereas (''a'', ''b'') is used for the ordered pair). If ''A'' and ''B'' are sets, then the Cartesian product (or simply product) is defined to be: :''A'' × ''B'' = . That is, ''A'' × ''B'' is the set of all ordered pairs whose first coordinate is an element of ''A'' and whose second coordinate is an element of ''B''. This definition may be extended to a set ''A'' × ''B'' × ''C'' of ordered triples, and more generally to sets of ordered n-tuples for any positive integer ''n''. It is even possible to define infinite
Cartesian product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s, but this requires a more recondite definition of the product. Cartesian products were first developed by
René Descartes René Descartes ( or ; ; Latinisation of names, Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French-born philosopher, Mathematics, mathematician, and scientist who spent a large portion of his working life in the Du ...

in the context of
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches ...
. If R denotes the set of all
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, then R2 := R × R represents the
Euclidean plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
and R3 := R × R × R represents three-dimensional
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
.

# Some important sets

There are some ubiquitous sets for which the notation is almost universal. Some of these are listed below. In the list, ''a'', ''b'', and ''c'' refer to
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s, and ''r'' and ''s'' are
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s. #
Natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s are used for counting. A
blackboard bold Image:Blackboard bold.svg, 250px, An example of blackboard bold letters Blackboard bold is a typeface style that is often used for certain symbols in mathematics, mathematical texts, in which certain lines of the symbol (usually vertical or near-v ...

capital N ($\mathbb$) often represents this set. #
Integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s appear as solutions for ''x'' in equations like ''x'' + ''a'' = ''b''. A blackboard bold capital Z ($\mathbb$) often represents this set (from the German ''Zahlen'', meaning ''numbers''). #
Rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s appear as solutions to equations like ''a'' + ''bx'' = ''c''. A blackboard bold capital Q ($\mathbb$) often represents this set (for ''
quotient In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne' ...
'', because R is used for the set of real numbers). #
Algebraic number An algebraic number is any complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, ev ...
s appear as solutions to
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

equations (with integer coefficients) and may involve radicals (including $i=\sqrt$) and certain other
irrational number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s. A Q with an overline ($\overline$) often represents this set. The overline denotes the operation of
algebraic closure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. #
Real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s represent the "real line" and include all numbers that can be approximated by rationals. These numbers may be rational or algebraic but may also be
transcendental number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, which cannot appear as solutions to polynomial equations with rational coefficients. A blackboard bold capital R ($\mathbb$) often represents this set. #
Complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s are sums of a real and an imaginary number: $r+s\,i$. Here either $r$ or $s$ (or both) can be zero; thus, the set of real numbers and the set of strictly imaginary numbers are subsets of the set of complex numbers, which form an
algebraic closure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
for the set of real numbers, meaning that every polynomial with coefficients in $\mathbb$ has at least one
root In vascular plant Vascular plants (from Latin ''vasculum'': duct), also known as Tracheophyta (the tracheophytes , from Greek τραχεῖα ἀρτηρία ''trācheia artēria'' 'windpipe' + φυτά ''phutá'' 'plants'), form a large grou ...
in this set. A blackboard bold capital C ($\mathbb$) often represents this set. Note that since a number $r+s\,i$ can be identified with a point $\left(r,s\right)$ in the plane, $\mathbb$ is basically "the same" as the
Cartesian product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
$\R\times\R$ ("the same" meaning that any point in one determines a unique point in the other and for the result of calculations, it doesn't matter which one is used for the calculation, as long as multiplication rule is appropriate for $\mathbb$).

# Paradoxes in early set theory

The unrestricted formation principle of sets referred to as the axiom schema of unrestricted comprehension, :''If'' ''is a property, then there exists a set'' (false), is the source of several early appearing paradoxes: * led, in the year 1897, to the
Burali-Forti paradox In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ty ...
, the first published
antinomy Antinomy (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as o ...
. * produced
Cantor's paradox In set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ar ...
in 1897. * yielded Cantor's second antinomy in the year 1899. Here the property is true for all , whatever may be, so would be a
universal set In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ty ...
, containing everything. *, i.e. the set of all sets that do not contain themselves as elements gave
Russell's paradox In the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theori ...

in 1902. If the axiom schema of unrestricted comprehension is weakened to the
axiom schema of specification In many popular versions of axiomatic set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected ...
or axiom schema of separation, :''If'' ''is a property, then for any set'' ''there exists a set'' , then all the above paradoxes disappear. There is a corollary. With the axiom schema of separation as an axiom of the theory, it follows, as a theorem of the theory: :''The set of all sets does not exist''. Or, more spectacularly (Halmos' phrasing): There is no
universe The universe ( la, universus) is all of space and time and their contents, including planets, stars, galaxy, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development ...
. ''Proof'': Suppose that it exists and call it . Now apply the axiom schema of separation with and for use . This leads to Russell's paradox again. Hence can't exist in this theory. Related to the above constructions is formation of the set *, where the statement following the implication certainly is false. It follows, from the definition of , using the usual inference rules (and some afterthought when reading the proof in the linked article below) both that and holds, hence . This is
Curry's paradox Curry's paradox is a paradox A paradox, also known as an antinomy, is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, ...
. It is (perhaps surprisingly) not the possibility of that is problematic. It is again the axiom schema of unrestricted comprehension allowing for . With the axiom schema of specification instead of unrestricted comprehension, the conclusion doesn't hold and, hence is not a logical consequence. Nonetheless, the possibility of is often removed explicitly or, e.g. in ZFC, implicitly, by demanding the
axiom of regularity In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every Empty set, non-empty Set (mathematics), set ''A'' contains an element that is Disjoint sets, disjoi ...
to hold. One consequence of it is :''There is no set'' ''for which'' , or, in other words, no set is an element of itself. The axiom schema of separation is simply too weak (while unrestricted comprehension is a very strong axiom—too strong for set theory) to develop set theory with its usual operations and constructions outlined above. The axiom of regularity is of a restrictive nature as well. Therefore, one is led to the formulation of other axioms to guarantee the existence of enough sets to form a set theory. Some of these have been described informally above and many others are possible. Not all conceivable axioms can be combined freely into consistent theories. For example, the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

of ZFC is incompatible with the conceivable every set of reals is
Lebesgue measurableIn measure theory In mathematics, a measure on a set (mathematics), set is a systematic way to assign a number, intuitively interpreted as its size, to some subsets of that set, called measurable sets. In this sense, a measure is a generalization ...
. The former implies the latter is false.

*
Algebra of sets In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
*
Axiomatic set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
*
Internal set theoryInternal set theory (IST) is a mathematical theory of Set (mathematics), sets developed by Edward Nelson that provides an axiomatic basis for a portion of the nonstandard analysis introduced by Abraham Robinson. Instead of adding new elements to the ...
*
List of set identities and relations This article lists mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and ...
*
Set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
*
Set (mathematics) In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
*
Partially ordered set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

# References

* Bourbaki, N., ''Elements of the History of Mathematics'', John Meldrum (trans.), Springer-Verlag, Berlin, Germany, 1994. * * Devlin, K.J., ''The Joy of Sets: Fundamentals of Contemporary Set Theory'', 2nd edition, Springer-Verlag, New York, NY, 1993. * María J. Frápolli, Frápolli, María J., 1991, "Is Cantorian set theory an iterative conception of set?". ''Modern Logic'', v. 1 n. 4, 1991, 302–318. * * ** ** * * Kelley, J.L., ''General Topology'', Van Nostrand Reinhold, New York, NY, 1955. * van Heijenoort, J., ''From Frege to Gödel, A Source Book in Mathematical Logic, 1879-1931'', Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977. . * * *