Naive set theory is one of several theories of sets used in the discussion of the
foundations of mathematicsFoundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
. The informal content of this
naive set theory supports both the aspects of mathematical sets familiar in
discrete mathematicsDiscrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. Real numbers and rational numbers have the property that between any two numbers a third can be found, and consequently these numbers vary "smoothly"...
(for example
Venn diagramVenn diagrams or set diagrams are diagrams that show all hypothetically possible logical relations between a finite collection of sets . Venn diagrams were conceived around 1880 by John Venn...
s and symbolic reasoning about their Boolean algebra), and the everyday usage of set theory concepts in most contemporary mathematics.
Sets are of great importance in
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
; in fact, in modern formal treatments, most mathematical objects (
numberA number is a mathematical object used in counting and measuring. A notational symbol which represents a number is called a numeral, but in common usage the word number is used for both the abstract object and the symbol, as well as for the word for the number...
s,
relationIn mathematics , a relation is a property that assigns truth values to combinations of k individuals. Typically, the property describes a possible connection between the components of a k-tuple...
s,
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
s, etc.) are defined in terms of sets. Naive set theory can be seen as a stepping-stone to more formal treatments, and suffices for many purposes.
Requirements
In the sense of this article, a
naive theory is a non-formalized theory, that is, a theory that uses a
natural languageIn the philosophy of language, a natural language is any language which arises in an unpremeditated fashion as the result of the innate facility for language possessed by the human intellect. A natural language is typically used for communication, and may be spoken, signed, or written...
to describe sets. The words
and,
or,
if ... then,
not,
for some,
for every are not subject to rigorous definition. It is useful to study sets naively at an early stage of mathematics in order to develop facility for working with them. Furthermore, a firm grasp of set theoretical concepts from a naive standpoint is important as a first stage in understanding the motivation for the formal axioms of set theory.
This article develops a naive theory. Sets are defined informally and a few of their properties are investigated. Links in this article to specific axioms of set theory describe some of the relationships between the informal discussion here and the formal axiomatization of set theory, but no attempt is made to justify every statement on such a basis. The first development of
set theoryThe modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
was a naive set theory. It was created at the end of the 19th century by
Georg CantorGeorg Ferdinand Ludwig Phillip Cantor was a German mathematician, born in Russia. He is best known as the creator of set theory, which has become a fundamental theory in mathematics...
as part of his study of
infinite setIn set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Some examples are:* the set of all integers, {..., -1, 0, 1, 2, ...}, is a countably infinite set; and...
s.
As it turned out, assuming that one can perform any operations on sets without restriction leads to
paradoxA paradox is a statement or group of statements that leads to a contradiction or a situation which defies intuition. The term is also used for an apparent contradiction that actually expresses a non-dual truth...
es such as
Russell's paradoxIn the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction....
and Berry's paradox. Some believe that
Georg CantorGeorg Ferdinand Ludwig Phillip Cantor was a German mathematician, born in Russia. He is best known as the creator of set theory, which has become a fundamental theory in mathematics...
's set theory was not actually implicated by these
paradoxA paradox is a statement or group of statements that leads to a contradiction or a situation which defies intuition. The term is also used for an apparent contradiction that actually expresses a non-dual truth...
es (see Frápolli 1991); one difficulty in determining this with certitude is that Cantor did not provide an axiomatization of his system. It is undisputed that, by 1900, Cantor was aware of some of the paradoxes and did not believe that they discredited his theory. Frege did explicitly axiomatize a theory, in which the formalized version of naive set theory can be interpreted, and it is this formal theory which
Bertrand RussellBertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was an English philosopher, logician, mathematician, historian, and social critic. Although he spent the majority of his life in England, he was born in Wales, where he also died.Russell led the British "revolt against idealism" in the...
actually addressed when he presented his paradox.
Axiomatic set theory was developed in response to these early attempts to study set theory, with the goal of determining precisely what operations were allowed and when. Today, when mathematicians talk about "set theory" as a field, they usually mean axiomatic set theory. Informal applications of set theory in other fields are sometimes referred to as applications of "naive set theory", but usually are understood to be justifiable in terms of an
axiomatic systemIn mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems...
(normally the
Zermelo–Fraenkel set theoryZermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics...
).
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 can be done by systematically making explicit all the axioms, as in the case of the well-known book
Naive Set Theory by
Paul HalmosPaul Richard Halmos was a Hungarian-born Jewish American mathematician who made fundamental advances in the areas of probability theory, statistics, operator theory, ergodic theory, functional analysis , and mathematical logic...
, which is actually a somewhat (not all that) informal presentation of the usual axiomatic
Zermelo–Fraenkel set theoryZermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics...
. It is 'naive' in that the language and notations are those of ordinary informal mathematics, and in that it doesn't deal with consistency or completeness of the axiom system. However, the term
naive set theory is 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; care is required to tell which sense is intended.
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
integerThe integers are natural numbers including 0 and their negatives . They are numbers that can be written without a fractional or decimal component, and fall within the set {.....
s. Clearly, the set of even numbers is infinitely large; there is no requirement that a set be finite.
If
x is a member of
A, then it is also said that
x belongs to A, or that
x is in
A. In this case, we write
x ∈
A.
(The symbol ∈ is a derivation from the
Greek letterThe Greek alphabet is a set of twenty-four letters that has been used to write the Greek language since the late 9th or early 8th century BCE. It is the first and oldest alphabet in the narrow sense that it notes each vowel and consonant with a separate symbol. It is as such in continuous use to...
epsilonEpsilon is the fifth letter of the Greek alphabet, corresponding phonetically to a close-mid front unrounded vowel /e/. In the system of Greek numerals it has a value of 5. It was derived from the Phoenician letter He...
, "ε", introduced by Peano in 1888.)
The symbol ∉ is sometimes used to write
x ∉
A, meaning "x is not in A".
Two sets
A and
B are defined to be
equalEquality, or more formally the identity relation, is the binary relation on a set X defined by .The identity relation is the paradigmatic example of the more general concept of an equivalence relation on a set: those binary relations which are reflexive, symmetric, and transitive. The relation of...
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 extensionalityIn 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 :...
.) 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 numberIn mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-six prime numbers are:An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The number 1 is by definition not a prime number...
s less than 6.
If the sets
A and
B are equal, this is denoted symbolically as
A =
B (as usual).
We also allow for an
empty setIn mathematics, and more specifically set theory, the empty set is the unique set having no members; its size is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
, often denoted Ø and sometimes : a set without any members at all.
Since a set is determined completely by its elements, there can only be one empty set. (See
axiom of empty setIn axiomatic set theory, the axiom of empty set is an axiom of Zermelo–Fraenkel set theory, the fragment thereof Burgess calls ST, and Kripke–Platek set theory.- Formal statement :In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:...
.) Note that Ø ≠ {Ø}.
Specifying sets
The simplest way to describe a set is to list its elements between curly braces (known as defining a set
extensionally). Thus {1,2} denotes the set whose only elements are 1 and 2.
(See
axiom of pairingIn axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo-Fraenkel set theory.- Formal statement :...
.)
Note the following points:
- Order of elements is immaterial; for example, {1,2} = {2,1}.
- Repetition (multiplicity) of elements is irrelevant; for example, {1,2,2} = {1,1,1,2} = {1,2}.
(These are consequences of the definition of equality in the previous section.)
This notation can be informally abused by saying something like {dogs} 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.
We can also use the notation {
x :
P(
x)}, or sometimes {
x |
P(
x)}, to denote the set containing all objects for which the condition
P holds (known as defining a set
intensionally).
For example, {
x :
x is a real number} denotes the set of
real numberIn mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
s, {
x :
x has blonde hair} denotes the set of everything with blonde hair, and {
x :
x is a dog} denotes the set of all dogs.
This notation is called
set-builder notationIn set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by stating the properties that its members must satisfy...
(or "
set comprehension", particularly in the context of
Functional programmingIn computer science, functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions and avoids state and mutable data. It emphasizes the application of functions, in contrast to the imperative programming style, which emphasizes changes in state...
).
Some variants of set builder notation are:
- {x ∈ A : P(x)} denotes the set of all x that are already members of A such that the condition P holds for x. For example, if Z is the set of integer
The integers are natural numbers including 0 and their negatives . They are numbers that can be written without a fractional or decimal component, and fall within the set {.....
s, then {x ∈ Z : x is even} is the set of all even-General:Even may refer to:* Even, a Scandinavian male personal name .* Even , an ethnic group from Siberia and Russian Far East**Even language, a language spoken by the Evens...
integers. (See axiom of specification.)
- {F(x) : x ∈ A} denotes the set of all objects obtained by putting members of the set A into the formula F. For example, {2x : x ∈ Z} is again the set of all even integers. (See axiom of replacement.)
- {F(x) : P(x)} is the most general form of set builder notation. For example, {xs owner : x is a dog} is the set of all dog owners.
Subsets
Given two sets A
and B
we say that A
is a subsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide...
of B
if every element of A
is also an element of B
.
Notice that in particular, B
is a subset of itself; a subset of B
that isn't 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. PresidentsThe President of the United States is the head of state and head of government of the United States and is the highest political official in the United States by influence and recognition...
.
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
.
Note that 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 proveIn mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
that two sets are equal, one aims to show these two inclusions. Note that the empty setIn mathematics, and more specifically set theory, the empty set is the unique set having no members; its size is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
is a subset of every set (the statement that all elements of the empty set are also members of any set A
is vacuously true).
The set of all subsets of a given set A
is called the power setIn mathematics, given a set S, the power set of S, written , P, ℘ or 2
S, is the set of all subsets of S. In axiomatic set theory In mathematics, given a set S, the power set (or powerset) of S, written , P(S), ℘(S) or 2
S, is the set of all subsets of S. In...
of A
and is denoted by or ; the "P
" is sometimes in a fancy font. If the set A
has n elements, then will have elements.
Universal sets and absolute complements
In certain contexts we may consider all sets under consideration as being subsets of some given
universal setIn mathematics, and particularly in set theory and the foundations of mathematics, a universe is a class that contains all the entities one wishes to consider in a given situation...
.
For instance, if we are investigating properties of the
real numberIn mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
s R
(and subsets of R
), then we may take R
as our universal set. A universal set is only temporarily defined by the context; there is no such thing as a "universal" universal set, "the set of everything" (see Paradoxes
below).
Given a universal set U
and a subset A of U
, we may define the complementIn 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 to another...
of A (in U
) as
- AC := {x
U
: x A}.
In other words, AC ("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, we may construct their
unionIn set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets gives a set .- Definition :A simple example:...
.
This is the set consisting of all objects which are elements of
A or of
B or of both (see
axiom of unionIn axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of union is one of the axioms of Zermelo-Fraenkel set theory, stating that, for any set x there is a set y whose elements are precisely the elements of the elements of x...
). It is denoted by
A ∪
B.
The
intersectionIn mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
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 complementIn 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 to another...
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 := {x : (x ∈ A) or
In logic and mathematics, or, also known as logical disjunction or inclusive disjunction, is a logical operator that results in true whenever one or more of its operands are true. E.g. in this context, "A or B" is true if A is true, or if B is true, or if both A and B are true. In grammar, or is a...
(x ∈ B)};
- A ∩ B := {x : (x ∈ A) and
In logic and mathematics, logical conjunction or and is a two-place logical connective that has the value true if both of its operands are true, otherwise a value of false.-Notation:...
(x ∈ B)} = {x ∈ A : x ∈ B} = {x ∈ B : x ∈ A};
- A \ B := {x : (x ∈ A) and not
In logic and mathematics, negation is an operation on propositions. For example, in classical logic negation is normally interpreted by the truth function that takes truth to falsity and vice versa...
(x ∈ B) } = {x ∈ A : not (x ∈ B)}.
Notice that
A doesn't have to be a subset of
B for
B \
A to make sense; this is the difference between the relative complement and the absolute complement 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 human being is over 1000 years old, so
E ∩
F must be the
empty setIn mathematics, and more specifically set theory, the empty set is the unique set having no members; its size is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
{}.
For any set
A, the power set is a Boolean algebra under the operations of union and intersection.
Ordered pairs and Cartesian products
Intuitively, an
ordered pairIn mathematics, an ordered pair is a collection of objects having two coordinates , such that one can always uniquely determine the object, which is the first coordinate of the pair as well as the second coordinate...
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), is 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 {a,b} with a
total orderIn mathematics and set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
.
(The notation (
a,
b) is also used to denote an open interval on the real number line, 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 productIn mathematics, a Cartesian product is the direct product of two sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to this concept....
(or simply
product) is defined to be:
- A × B = {(a,b) : a is in A and b is in 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.
We can extend this definition 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 productIn mathematics, a Cartesian product is the direct product of two sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to this concept....
s, but to do this we need a more recondite definition of the product.
Cartesian products were first developed by
René DescartesRené Descartes , , also known as Renatus Cartesius , was a French philosopher, mathematician, physicist, and writer who spent most of his adult life in the Dutch Republic...
in the context of
analytic geometryAnalytic geometry, also known as coordinate geometry, analytical geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis...
.
If
R denotes the set of all
real numberIn mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
s, then
R2 :=
R ×
R represents the Euclidean plane and
R3 :=
R ×
R ×
R represents three-dimensional
Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions higher dimensions...
.
Some important sets
Note: In this section, a, b, and c are natural numberIn mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {, , , ...} according to the traditional definition or the set of non-negative integers {, 1, 2, ...} according to...
s, and r and s are real numberIn mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
s.
- Natural number
In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {, , , ...} according to the traditional definition or the set of non-negative integers {, 1, 2, ...} according to...
s are used for counting. A blackboard boldBlackboard bold is a typeface style often used for certain symbols in mathematics and physics texts, in which certain lines of the symbol are doubled. The symbols usually denote number sets. Blackboard bold symbols are also referred to as double struck, although they cannot actually be produced by...
capital N often represents this set.
- Integer
The integers are natural numbers including 0 and their negatives . They are numbers that can be written without a fractional or decimal component, and fall within the set {.....
s appear as solutions for x in equations like x + a = b. A blackboard bold capital Z often represents this set (from the German Zahlen, meaning numbers).
- Rational number
In mathematics, a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer corresponds to a rational number. The set of all rational numbers is usually denoted .Formally each rational...
s appear as solutions to equations like a + bx = c. A blackboard bold capital Q often represents this set (for quotientIn mathematics, a quotient is the result of a division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient can also be expressed as the number of times the divisor divides into the dividend....
, because R is used for the set of real numbers).
- Algebraic number
In mathematics, an algebraic number is a complex number that is a root of a non-zero polynomial in one variable with rational coefficients...
s appear as solutions to polynomialIn mathematics, a polynomial is a finite length expression constructed from variables and constants, by using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents...
equations (with integer coefficients) and may involve radicalIn mathematics, a root of a number x is any number which, when repeatedly multiplied by itself, eventually yields x:In terms of exponentiation, r is a root of x iffor some positive integer n...
s and certain other irrational numberIn 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 m/n, where m and n are integers, with n non-zero. Informally, this means numbers that cannot be represented as simple fractions...
s. A blackboard bold capital A or a Q with an overline often represents this set. The overline denotes the operation of algebraic closureIn mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
.
- Real number
In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
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 numberIn mathematics, a transcendental number is a number that is not algebraic, that is, not a solution of a non-constant polynomial equation with rational coefficients....
s, which cannot appear as solutions to polynomial equations with rational coefficients. A blackboard bold capital R often represents this set.
- Complex number
A complex number, in mathematics, is a number comprising a real number and an imaginary number; it can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit, having the property that i
2 = −1...
s are sums of a real and an imaginary number: r + si. Here both r and s can equal zero; thus, the set of real numbers and the set of imaginary numbers are subsets of the set of complex numbers, which form an algebraic closureIn mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
for the set of real numbers, meaning that every polynomial with coefficients in has at least one root in this set. A blackboard bold capital C often represents this set. Note that since a number r + si can be identified with a point (r, s) in the plane, C is basically "the same" as the Cartesian product R×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).
See also
- Algebra of sets
The algebra of sets develops and describes the basic properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion...
- Axiomatic set theory
- Internal set theory
Internal set theory is a mathematical theory of sets developed by Edward Nelson which provides an axiomatic basis for a portion of the non-standard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, the axioms introduce a new term, "standard", which can...
- Set theory
The modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
- Set (mathematics)
- Partially ordered set
In mathematics, especially order theory, a partially ordered set formalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that describes, for certain pairs of elements in the set, the requirement...
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