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In mathematics, a set is a collection of different things; these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. A set may be finite or infinite, depending whether the number of its elements is finite or not. There is a unique set with no elements, called the empty set; a set with a single element is a singleton.
![image](https://www.english.nina.az/wikipedia/image/aHR0cHM6Ly93d3cuZW5nbGlzaC5uaW5hLmF6L3dpa2lwZWRpYS9pbWFnZS9hSFIwY0hNNkx5OTFjR3h2WVdRdWQybHJhVzFsWkdsaExtOXlaeTkzYVd0cGNHVmthV0V2WTI5dGJXOXVjeTkwYUhWdFlpOHpMek0zTDBWNFlXMXdiR1ZmYjJaZllWOXpaWFF1YzNabkx6SXlNSEI0TFVWNFlXMXdiR1ZmYjJaZllWOXpaWFF1YzNabkxuQnVadz09LnBuZw==.png)
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Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century.
Context
Before the end of the 19th century, sets were not studied specifically, and were not clearly distinguished from sequences. Most mathematicians considered infinity as potential—meaning that it is the result of an endless process—and were reluctant to consider infinite sets, that is sets whose number of members is not a natural number. In particular, a line was not consideed as the set of its points, but as a locus where points may be located.
The mathematical study of sets began with Georg Cantor (1845–1918). This provided some counterintuitive facts and paradoxes. For example, the number line has an infinite number of elements that is strictly larger than the infinite number of natural numbers, and any line segment has the same number of elements as the whole space. Also, Russel's paradox implies that the phrase "the set of all sets" is self-contradictory.
Together with other counterintuitive results, this led to the foundational crisis of mathematics, which was eventually resolved with the general adoption of Zermelo–Fraenkel set theory as a robust foundation of set theory and all mathematics.
Meanwhile, sets started to be widely used in all mathematics. In particular, algebraic structures and mathematical spaces are typically defined in terms of sets. Also, many older mathematical results are restated in terms of sets. For example, Euclid's theorem is often stated as "the set of the prime numbers is infinite". This wide use of sets in mathematics was prophesied by David Hilbert when saying: "No one will drive us from the paradise which Cantor created for us." However, he not imagined probably that sets are nowadays taught in the first grades of mathematics.
Generally, the common usage of sets in mathematics does not requires the full power of Zermelo–Fraenkel set theory. In mathematical practice, sets can be manipulated independently of the logical framework of this theory.
The object of this article is to summarize the manipulation rules and properties of sets that are commonly used in mathematics, without reference to any logical framework.
Definition
In mathematics, a set is a collection of different things. These things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, functions, or even other sets. A set may also be called a collection or family, especially when its elements are themselves sets; this may avoid the confusion between the set and its members, and may make reading easier. The elements of a well-defined collection constitute a set, such is the case for the set of prime numbers and the set of all students in a class.
If is an element of a set
, one says that
belongs to
or is in
, and this is written as
. The statement "
is not in
" is written as
, which can also be read as "y is not in B". For example, if
is the set of the integers, one has
and
.
Each set is uniquely characterized by its elements. In particular, two sets that have precisely the same elements are equal (they are the same set). This property, called extensionality, can be written in formula as
This implies that there is only one set with no element, the empty set (or null set) that is denoted , or
A set is finite if there exists a natural number such that the
first natural numbers can be put in one to one correspondence with the elements of the set. In this case, one says that
is the number of elements of the set. A set is infinite if such an
does not exists. The empty set is a finite set with
elements.
The natural numbers form an infinite set, commonly denoted .
A singleton is a set with exactly one element. If is this element, the singleton is denoted
If
is itself a set, it must not be confused with
For example,
is a set with no elements, while
is a singleton with
as its unique element.
Roster notation
Roster or enumeration notation defines a set by listing its elements between curly brackets, separated by commas:
This notation was introduced by Ernst Zermelo in 1908. In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant in contrast to a sequence or a tuple, where the ordering of the terms matters. For example, {2, 4, 6} and {4, 6, 4, 2} represent the same set.
For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis '...'. For instance, the set of the first thousand positive integers may be specified in roster notation as
Infinite sets in roster notation
An infinite set is a set with an infinite number of elements. If the pattern of its elements is obvious, an infinite set can be given in roster notation, with an ellipsis placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is
- {0, 1, 2, 3, 4, ...},
and the set of all integers is
- {..., −3, −2, −1, 0, 1, 2, 3, ...}.
Set-builder notation
Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. For example, a set F can be defined as follows:
In this notation, the vertical bar "|" means "such that", and the description can be interpreted as "F is the set of all numbers n such that n is an integer in the range from 0 to 19 inclusive". Some authors use a colon ":" instead of the vertical bar.
Subsets
If every element of set A is also in B, then A is described as being a subset of B, or contained in B, written A ⊆ B, or B ⊇ A. The latter notation may be read B contains A, B includes A, or B is a superset of A. The relationship between sets established by ⊆ is called inclusion or containment. Two sets are equal if they contain each other: A ⊆ B and B ⊆ A is equivalent to A = B.
If A is a subset of B, but A is not equal to B, then A is called a proper subset of B. This can be written A ⊊ B. Likewise, B ⊋ A means B is a proper superset of A, i.e. B contains A, and is not equal to A.
A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use A ⊂ B and B ⊃ A to mean A is any subset of B (and not necessarily a proper subset), while others reserve A ⊂ B and B ⊃ A for cases where A is a proper subset of B.
Examples:
- The set of all humans is a proper subset of the set of all mammals.
- {1, 3} ⊂ {1, 2, 3, 4}.
- {1, 2, 3, 4} ⊆ {1, 2, 3, 4}.
The empty set is a subset of every set, and every set is a subset of itself:
- ∅ ⊆ A.
- A ⊆ A.
Euler and Venn diagrams
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B is a superset of A.
An Euler diagram is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If A is a subset of B, then the region representing A is completely inside the region representing B. If two sets have no elements in common, the regions do not overlap.
A Venn diagram, in contrast, is a graphical representation of n sets in which the n loops divide the plane into 2n zones such that for each way of selecting some of the n sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are A, B, and C, there should be a zone for the elements that are inside A and C and outside B (even if such elements do not exist).
Special sets of numbers in mathematics
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There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them.
Many of these important sets are represented in mathematical texts using bold (e.g. ) or blackboard bold (e.g.
) typeface. These include
or
, the set of all natural numbers:
(often, authors exclude 0);
or
, the set of all integers (whether positive, negative or zero):
;
or
, the set of all rational numbers (that is, the set of all proper and improper fractions):
. For example, −7/4 ∈ Q and 5 = 5/1 ∈ Q;
or
, the set of all real numbers, including all rational numbers and all irrational numbers (which include algebraic numbers such as
that cannot be rewritten as fractions, as well as transcendental numbers such as π and e);
or
, the set of all complex numbers: C = {a + bi | a, b ∈ R}, for example, 1 + 2i ∈ C.
Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it.
Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, represents the set of positive rational numbers.
Functions
A function (or mapping) from a set A to a set B is a rule that assigns to each "input" element of A an "output" that is an element of B; more formally, a function is a special kind of relation, one that relates each element of A to exactly one element of B. A function is called
- injective (or one-to-one) if it maps any two different elements of A to different elements of B,
- surjective (or onto) if for every element of B, there is at least one element of A that maps to it, and
- bijective (or a one-to-one correspondence) if the function is both injective and surjective — in this case, each element of A is paired with a unique element of B, and each element of B is paired with a unique element of A, so that there are no unpaired elements.
An injective function is called an injection, a surjective function is called a surjection, and a bijective function is called a bijection or one-to-one correspondence.
Cardinality
The cardinality of a set S, denoted |S|, is the number of members of S. For example, if B = {blue, white, red}, then |B| = 3. Repeated members in roster notation are not counted, so |{blue, white, red, blue, white}| = 3, too.
More formally, two sets share the same cardinality if there exists a bijection between them.
The cardinality of the empty set is zero.
Infinite sets and infinite cardinality
The list of elements of some sets is endless, or infinite. For example, the set of natural numbers is infinite. In fact, all the special sets of numbers mentioned in the section above are infinite. Infinite sets have infinite cardinality.
Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that the set of real numbers has greater cardinality than the set of natural numbers. Sets with cardinality less than or equal to that of are called countable sets; these are either finite sets or countably infinite sets (sets of the same cardinality as
); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of
are called uncountable sets.
However, it can be shown that the cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.
The continuum hypothesis
The continuum hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and the cardinality of a straight line. In 1963, Paul Cohen proved that the continuum hypothesis is independent of the axiom system ZFC consisting of Zermelo–Fraenkel set theory with the axiom of choice. (ZFC is the most widely-studied version of axiomatic set theory.)
Power sets
The power set of a set S is the set of all subsets of S. The empty set and S itself are elements of the power set of S, because these are both subsets of S. For example, the power set of {1, 2, 3} is {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}. The power set of a set S is commonly written as P(S) or 2S.
If S has n elements, then P(S) has 2n elements. For example, {1, 2, 3} has three elements, and its power set has 23 = 8 elements, as shown above.
If S is infinite (whether countable or uncountable), then P(S) is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of S with the elements of P(S) will leave some elements of P(S) unpaired. (There is never a bijection from S onto P(S).)
Partitions
A partition of a set S is a set of nonempty subsets of S, such that every element x in S is in exactly one of these subsets. That is, the subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is S.
Basic operations
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Suppose that a universal set U (a set containing all elements being discussed) has been fixed, and that A is a subset of U.
- The complement of A is the set of all elements (of U) that do not belong to A. It may be denoted Ac or A′. In set-builder notation,
. The complement may also be called the absolute complement to distinguish it from the relative complement below. Example: If the universal set is taken to be the set of integers, then the complement of the set of even integers is the set of odd integers.
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Given any two sets A and B,
- their union A ∪ B is the set of all things that are members of A or B or both.
- their intersection A ∩ B is the set of all things that are members of both A and B. If A ∩ B = ∅, then A and B are said to be disjoint.
- the set difference A \ B (also written A − B) is the set of all things that belong to A but not B. Especially when B is a subset of A, it is also called the relative complement of B in A. With Bc as the absolute complement of B (in the universal set U), A \ B = A ∩ Bc .
- their symmetric difference A Δ B is the set of all things that belong to A or B but not both. One has
.
- their cartesian product A × B is the set of all ordered pairs (a,b) such that a is an element of A and b is an element of B.
Examples:
- {1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}.
- {1, 2, 3} ∩ {3, 4, 5} = {3}.
- {1, 2, 3} − {3, 4, 5} = {1, 2}.
- {1, 2, 3} Δ {3, 4, 5} = {1, 2, 4, 5}.
- {a, b} × {1, 2, 3} = {(a,1), (a,2), (a,3), (b,1), (b,2), (b,3)}.
The operations above satisfy many identities. For example, one of De Morgan's laws states that (A ∪ B)′ = A′ ∩ B′ (that is, the elements outside the union of A and B are the elements that are outside A and outside B).
The cardinality of A × B is the product of the cardinalities of A and B. This is an elementary fact when A and B are finite. When one or both are infinite, multiplication of cardinal numbers is defined to make this true.
The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring and intersection as the multiplication of the ring.
Applications
Sets are ubiquitous in modern mathematics. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.
One of the main applications of naive set theory is in the construction of relations. A relation from a domain A to a codomain B is a subset of the Cartesian product A × B. For example, considering the set S = {rock, paper, scissors} of shapes in the game of the same name, the relation "beats" from S to S is the set B = {(scissors,paper), (paper,rock), (rock,scissors)}; thus x beats y in the game if the pair (x,y) is a member of B. Another example is the set F of all pairs (x, x2), where x is real. This relation is a subset of R × R, because the set of all squares is subset of the set of all real numbers. Since for every x in R, one and only one pair (x,...) is found in F, it is called a function. In functional notation, this relation can be written as F(x) = x2.
Principle of inclusion and exclusion
![image](https://www.english.nina.az/wikipedia/image/aHR0cHM6Ly93d3cuZW5nbGlzaC5uaW5hLmF6L3dpa2lwZWRpYS9pbWFnZS9hSFIwY0hNNkx5OTFjR3h2WVdRdWQybHJhVzFsWkdsaExtOXlaeTkzYVd0cGNHVmthV0V2WTI5dGJXOXVjeTkwYUhWdFlpODRMemcyTDBGZmRXNXBiMjVmUWk1emRtY3ZNakl3Y0hndFFWOTFibWx2Ymw5Q0xuTjJaeTV3Ym1jPS5wbmc=.png)
The inclusion–exclusion principle is a technique for counting the elements in a union of two finite sets in terms of the sizes of the two sets and their intersection. It can be expressed symbolically as
A more general form of the principle gives the cardinality of any finite union of finite sets:
History
The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, Menge, was coined by Bernard Bolzano in his work Paradoxes of the Infinite.
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Georg Cantor, one of the founders of set theory, gave the following definition at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:
A set is a gathering together into a whole of definite, distinct objects of our perception or our thought—which are called elements of the set.
Bertrand Russell introduced the distinction between a set and a class (a set is a class, but some classes, such as the class of all sets, are not sets; see Russell's paradox):
When mathematicians deal with what they call a manifold, aggregate, Menge, ensemble, or some equivalent name, it is common, especially where the number of terms involved is finite, to regard the object in question (which is in fact a class) as defined by the enumeration of its terms, and as consisting possibly of a single term, which in that case is the class.
Naive set theory
The foremost property of a set is that it can have elements, also called members. Two sets are equal when they have the same elements. More precisely, sets A and B are equal if every element of A is an element of B, and every element of B is an element of A; this property is called the extensionality of sets. As a consequence, e.g. {2, 4, 6} and {4, 6, 4, 2} represent the same set. Unlike sets, multisets can be distinguished by the number of occurrences of an element; e.g. [2, 4, 6] and [4, 6, 4, 2] represent different multisets, while [2, 4, 6] and [6, 4, 2] are equal. Tuples can even be distinguished by element order; e.g. (2, 4, 6) and (6, 4, 2) represent different tuples.
The simple concept of a set has proved enormously useful in mathematics, but paradoxes arise if no restrictions are placed on how sets can be constructed:
- Russell's paradox shows that the "set of all sets that do not contain themselves", i.e., {x | x is a set and x ∉ x}, cannot exist.
- Cantor's paradox shows that "the set of all sets" cannot exist.
Naïve set theory defines a set as any well-defined collection of distinct elements, but problems arise from the vagueness of the term well-defined.
Axiomatic set theory
In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve set theory, the properties of sets have been defined by axioms. Axiomatic set theory takes the concept of a set as a primitive notion. The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular mathematical propositions (statements) about sets, using first-order logic. According to Gödel's incompleteness theorems however, it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from paradox.
See also
- Algebra of sets
- Alternative set theory
- Category of sets
- Class (set theory)
- Family of sets
- Fuzzy set
- Mereology
- Principia Mathematica
Notes
- Some typographical variants are occasionally used, such as ϕ, or ϕ.
- The term unit set is also occasionally used.
References
- Cantor, Georg; Jourdain, Philip E.B. (Translator) (1915). Contributions to the founding of the theory of transfinite numbers. New York Dover Publications (1954 English translation).
By an 'aggregate' (Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Ganzen) M of definite and separate objects m of our intuition or our thought.
Here: p.85 - P. K. Jain; Khalil Ahmad; Om P. Ahuja (1995). Functional Analysis. New Age International. p. 1. ISBN 978-81-224-0801-0.
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- Thomas H. Cormen; Charles E Leiserson; Ronald L Rivest; Clifford Stein (2001). Introduction To Algorithms. MIT Press. p. 1070. ISBN 978-0-262-03293-3.
- Halmos 1960, p. 1.
- Hilbert, David (1926), "Über das Unendliche", Mathematische Annalen, vol. 95, pp. 161–190, doi:10.1007/BF01206605, JFM 51.0044.02, S2CID 121888793
- "Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können."
- Translated in Van Heijenoort, Jean, On the infinite, Harvard University Press
- Maddocks, J. R. (2004). Lerner, K. Lee; Lerner, Brenda Wilmoth (eds.). The Gale Encyclopedia of Science. Gale. pp. 3587–3589. ISBN 0-7876-7559-8.
- Devlin, Keith J. (1981). Sets, Functions and Logic: Basic concepts of university mathematics. Springer. pp. 32–33. ISBN 978-0-412-22660-1.
- Halmos 1960, p. 2.
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- "Set Symbols". www.mathsisfun.com. Retrieved 2020-08-19.
- Stoll, Robert (1974). Sets, Logic and Axiomatic Theories. W. H. Freeman and Company. pp. 5. ISBN 9780716704577.
- Aggarwal, M.L. (2021). "1. Sets". Understanding ISC Mathematics Class XI. Vol. 1. Arya Publications (Avichal Publishing Company). p. A=3.
- Sourendra Nath, De (January 2015). "Unit-1 Sets and Functions: 1. Set Theory". Chhaya Ganit (Ekadash Shreni). Scholar Books Pvt. Ltd. p. 5.
- Halmos 1960, p. 8.
- K.T. Leung; Doris Lai-chue Chen (1 July 1992). Elementary Set Theory, Part I/II. Hong Kong University Press. p. 27. ISBN 978-962-209-026-2.
- Charles Roberts (24 June 2009). Introduction to Mathematical Proofs: A Transition. CRC Press. p. 45. ISBN 978-1-4200-6956-3.
- David Johnson; David B. Johnson; Thomas A. Mowry (June 2004). Finite Mathematics: Practical Applications (Docutech Version). W. H. Freeman. p. 220. ISBN 978-0-7167-6297-3.
- Ignacio Bello; Anton Kaul; Jack R. Britton (29 January 2013). Topics in Contemporary Mathematics. Cengage Learning. p. 47. ISBN 978-1-133-10742-2.
- Susanna S. Epp (4 August 2010). Discrete Mathematics with Applications. Cengage Learning. p. 13. ISBN 978-0-495-39132-6.
- A. Kanamori, "The Empty Set, the Singleton, and the Ordered Pair", p.278. Bulletin of Symbolic Logic vol. 9, no. 3, (2003). Accessed 21 August 2023.
- Stephen B. Maurer; Anthony Ralston (21 January 2005). Discrete Algorithmic Mathematics. CRC Press. p. 11. ISBN 978-1-4398-6375-6.
- "Introduction to Sets". www.mathsisfun.com. Retrieved 2020-08-19.
- D. Van Dalen; H. C. Doets; H. De Swart (9 May 2014). Sets: Naïve, Axiomatic and Applied: A Basic Compendium with Exercises for Use in Set Theory for Non Logicians, Working and Teaching Mathematicians and Students. Elsevier Science. p. 1. ISBN 978-1-4831-5039-0.
- Alfred Basta; Stephan DeLong; Nadine Basta (1 January 2013). Mathematics for Information Technology. Cengage Learning. p. 3. ISBN 978-1-285-60843-3.
- Laura Bracken; Ed Miller (15 February 2013). Elementary Algebra. Cengage Learning. p. 36. ISBN 978-0-618-95134-5.
- Frank Ruda (6 October 2011). Hegel's Rabble: An Investigation into Hegel's Philosophy of Right. Bloomsbury Publishing. p. 151. ISBN 978-1-4411-7413-0.
- John F. Lucas (1990). Introduction to Abstract Mathematics. Rowman & Littlefield. p. 108. ISBN 978-0-912675-73-2.
- Weisstein, Eric W. "Set". Wolfram MathWorld. Retrieved 2020-08-19.
- Ralph C. Steinlage (1987). College Algebra. West Publishing Company. ISBN 978-0-314-29531-6.
- Felix Hausdorff (2005). Set Theory. American Mathematical Soc. p. 30. ISBN 978-0-8218-3835-8.
- Peter Comninos (6 April 2010). Mathematical and Computer Programming Techniques for Computer Graphics. Springer Science & Business Media. p. 7. ISBN 978-1-84628-292-8.
- Halmos 1960, p. 3.
- George Tourlakis (13 February 2003). Lectures in Logic and Set Theory: Volume 2, Set Theory. Cambridge University Press. p. 137. ISBN 978-1-139-43943-5.
- Yiannis N. Moschovakis (1994). Notes on Set Theory. Springer Science & Business Media. ISBN 978-3-540-94180-4.
- Arthur Charles Fleck (2001). Formal Models of Computation: The Ultimate Limits of Computing. World Scientific. p. 3. ISBN 978-981-02-4500-9.
- William Johnston (25 September 2015). The Lebesgue Integral for Undergraduates. The Mathematical Association of America. p. 7. ISBN 978-1-939512-07-9.
- Karl J. Smith (7 January 2008). Mathematics: Its Power and Utility. Cengage Learning. p. 401. ISBN 978-0-495-38913-2.
- John Stillwell (16 October 2013). The Real Numbers: An Introduction to Set Theory and Analysis. Springer Science & Business Media. ISBN 978-3-319-01577-4.
- David Tall (11 April 2006). Advanced Mathematical Thinking. Springer Science & Business Media. p. 211. ISBN 978-0-306-47203-9.
- Cantor, Georg (1878). "Ein Beitrag zur Mannigfaltigkeitslehre". Journal für die Reine und Angewandte Mathematik. 1878 (84): 242–258. doi:10.1515/crll.1878.84.242 (inactive 1 November 2024).
{{cite journal}}
: CS1 maint: DOI inactive as of November 2024 (link) - Cohen, Paul J. (December 15, 1963). "The Independence of the Continuum Hypothesis". Proceedings of the National Academy of Sciences of the United States of America. 50 (6): 1143–1148. Bibcode:1963PNAS...50.1143C. doi:10.1073/pnas.50.6.1143. JSTOR 71858. PMC 221287. PMID 16578557.
- Halmos 1960, p. 19.
- Halmos 1960, p. 20.
- Edward B. Burger; Michael Starbird (18 August 2004). The Heart of Mathematics: An invitation to effective thinking. Springer Science & Business Media. p. 183. ISBN 978-1-931914-41-3.
- Toufik Mansour (27 July 2012). Combinatorics of Set Partitions. CRC Press. ISBN 978-1-4398-6333-6.
- Halmos 1960, p. 28.
- José Ferreirós (16 August 2007). Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Birkhäuser Basel. ISBN 978-3-7643-8349-7.
- Steve Russ (9 December 2004). The Mathematical Works of Bernard Bolzano. OUP Oxford. ISBN 978-0-19-151370-1.
- William Ewald; William Bragg Ewald (1996). From Kant to Hilbert Volume 1: A Source Book in the Foundations of Mathematics. OUP Oxford. p. 249. ISBN 978-0-19-850535-8.
- Paul Rusnock; Jan Sebestík (25 April 2019). Bernard Bolzano: His Life and Work. OUP Oxford. p. 430. ISBN 978-0-19-255683-7.
- Georg Cantor (Nov 1895). "Beiträge zur Begründung der transfiniten Mengenlehre (1)". Mathematische Annalen (in German). 46 (4): 481–512.
- Bertrand Russell (1903) The Principles of Mathematics, chapter VI: Classes
- Jose Ferreiros (1 November 2001). Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Springer Science & Business Media. ISBN 978-3-7643-5749-8.
- Raatikainen, Panu (2022). Zalta, Edward N. (ed.). "Gödel's Incompleteness Theorems". Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Retrieved 2024-06-03.
References
- Dauben, Joseph W. (1979). Georg Cantor: His Mathematics and Philosophy of the Infinite. Boston: Harvard University Press. ISBN 0-691-02447-2.
- Halmos, Paul R. (1960). Naive Set Theory. Princeton, N.J.: Van Nostrand. ISBN 0-387-90092-6.
- Stoll, Robert R. (1979). Set Theory and Logic. Mineola, N.Y.: Dover Publications. ISBN 0-486-63829-4.
- Velleman, Daniel (2006). How To Prove It: A Structured Approach. Cambridge University Press. ISBN 0-521-67599-5.
External links
The dictionary definition of set at Wiktionary
- Cantor's "Beiträge zur Begründung der transfiniten Mengenlehre" (in German)
In mathematics a set is a collection of different things these things are called elements or members of the set and are typically mathematical objects of any kind numbers symbols points in space lines other geometrical shapes variables or even other sets A set may be finite or infinite depending whether the number of its elements is finite or not There is a unique set with no elements called the empty set a set with a single element is a singleton A set of polygons in an Euler diagramThis set equals the one depicted above since both have the very same elements Sets are ubiquitous in modern mathematics Indeed set theory more specifically Zermelo Fraenkel set theory has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century ContextBefore the end of the 19th century sets were not studied specifically and were not clearly distinguished from sequences Most mathematicians considered infinity as potential meaning that it is the result of an endless process and were reluctant to consider infinite sets that is sets whose number of members is not a natural number In particular a line was not consideed as the set of its points but as a locus where points may be located The mathematical study of sets began with Georg Cantor 1845 1918 This provided some counterintuitive facts and paradoxes For example the number line has an infinite number of elements that is strictly larger than the infinite number of natural numbers and any line segment has the same number of elements as the whole space Also Russel s paradox implies that the phrase the set of all sets is self contradictory Together with other counterintuitive results this led to the foundational crisis of mathematics which was eventually resolved with the general adoption of Zermelo Fraenkel set theory as a robust foundation of set theory and all mathematics Meanwhile sets started to be widely used in all mathematics In particular algebraic structures and mathematical spaces are typically defined in terms of sets Also many older mathematical results are restated in terms of sets For example Euclid s theorem is often stated as the set of the prime numbers is infinite This wide use of sets in mathematics was prophesied by David Hilbert when saying No one will drive us from the paradise which Cantor created for us However he not imagined probably that sets are nowadays taught in the first grades of mathematics Generally the common usage of sets in mathematics does not requires the full power of Zermelo Fraenkel set theory In mathematical practice sets can be manipulated independently of the logical framework of this theory The object of this article is to summarize the manipulation rules and properties of sets that are commonly used in mathematics without reference to any logical framework DefinitionIn mathematics a set is a collection of different things These things are called elements or members of the set and are typically mathematical objects of any kind numbers symbols points in space lines other geometrical shapes variables functions or even other sets A set may also be called a collection or family especially when its elements are themselves sets this may avoid the confusion between the set and its members and may make reading easier The elements of a well defined collection constitute a set such is the case for the set of prime numbers and the set of all students in a class If x displaystyle x is an element of a set S displaystyle S one says that x displaystyle x belongs to S displaystyle S or is in S displaystyle S and this is written as x S displaystyle x in S The statement y displaystyle y is not in S displaystyle S is written as y S displaystyle y not in S which can also be read as y is not in B For example if Z displaystyle mathbb Z is the set of the integers one has 3 Z displaystyle 3 in mathbb Z and 1 5 Z displaystyle 1 5 not in mathbb Z Each set is uniquely characterized by its elements In particular two sets that have precisely the same elements are equal they are the same set This property called extensionality can be written in formula as A B x x A x B displaystyle A B iff forall x x in A iff x in B This implies that there is only one set with no element the empty set or null set that is denoted displaystyle varnothing emptyset or displaystyle A set is finite if there exists a natural number n displaystyle n such that the n displaystyle n first natural numbers can be put in one to one correspondence with the elements of the set In this case one says that n displaystyle n is the number of elements of the set A set is infinite if such an n displaystyle n does not exists The empty set is a finite set with 0 displaystyle 0 elements The natural numbers form an infinite set commonly denoted N displaystyle mathbb N A singleton is a set with exactly one element If x displaystyle x is this element the singleton is denoted x displaystyle x If x displaystyle x is itself a set it must not be confused with x displaystyle x For example displaystyle emptyset is a set with no elements while displaystyle emptyset is a singleton with displaystyle emptyset as its unique element Roster notation Roster or enumeration notation defines a set by listing its elements between curly brackets separated by commas A 4 2 1 3 B blue white red This notation was introduced by Ernst Zermelo in 1908 In a set all that matters is whether each element is in it or not so the ordering of the elements in roster notation is irrelevant in contrast to a sequence or a tuple where the ordering of the terms matters For example 2 4 6 and 4 6 4 2 represent the same set For sets with many elements especially those following an implicit pattern the list of members can be abbreviated using an ellipsis For instance the set of the first thousand positive integers may be specified in roster notation as 1 2 3 1000 Infinite sets in roster notation An infinite set is a set with an infinite number of elements If the pattern of its elements is obvious an infinite set can be given in roster notation with an ellipsis placed at the end of the list or at both ends to indicate that the list continues forever For example the set of nonnegative integers is 0 1 2 3 4 and the set of all integers is 3 2 1 0 1 2 3 Set builder notation Set builder notation specifies a set as a selection from a larger set determined by a condition on the elements For example a set F can be defined as follows F n n is an integer and 0 n 19 displaystyle F n mid n text is an integer and 0 leq n leq 19 In this notation the vertical bar means such that and the description can be interpreted as F is the set of all numbers n such that n is an integer in the range from 0 to 19 inclusive Some authors use a colon instead of the vertical bar SubsetsIf every element of set A is also in B then A is described as being a subset of B or contained in B written A B or B A The latter notation may be read B contains A B includes A or B is a superset of A The relationship between sets established by is called inclusion or containment Two sets are equal if they contain each other A B and B A is equivalent to A B If A is a subset of B but A is not equal to B then A is called a proper subset of B This can be written A B Likewise B A means B is a proper superset of A i e B contains A and is not equal to A A third pair of operators and are used differently by different authors some authors use A B and B A to mean A is any subset of B and not necessarily a proper subset while others reserve A B and B A for cases where A is a proper subset of B Examples The set of all humans is a proper subset of the set of all mammals 1 3 1 2 3 4 1 2 3 4 1 2 3 4 The empty set is a subset of every set and every set is a subset of itself A A A Euler and Venn diagramsA is a subset of B B is a superset of A An Euler diagram is a graphical representation of a collection of sets each set is depicted as a planar region enclosed by a loop with its elements inside If A is a subset of B then the region representing A is completely inside the region representing B If two sets have no elements in common the regions do not overlap A Venn diagram in contrast is a graphical representation of n sets in which the n loops divide the plane into 2n zones such that for each way of selecting some of the n sets possibly all or none there is a zone for the elements that belong to all the selected sets and none of the others For example if the sets are A B and C there should be a zone for the elements that are inside A and C and outside B even if such elements do not exist Special sets of numbers in mathematicsThe natural numbers N displaystyle mathbb N are contained in the integers Z displaystyle mathbb Z which are contained in the rational numbers Q displaystyle mathbb Q which are contained in the real numbers R displaystyle mathbb R which are contained in the complex numbers C displaystyle mathbb C There are sets of such mathematical importance to which mathematicians refer so frequently that they have acquired special names and notational conventions to identify them Many of these important sets are represented in mathematical texts using bold e g Z displaystyle mathbf Z or blackboard bold e g Z displaystyle mathbb Z typeface These include N displaystyle mathbf N or N displaystyle mathbb N the set of all natural numbers N 0 1 2 3 displaystyle mathbf N 0 1 2 3 often authors exclude 0 Z displaystyle mathbf Z or Z displaystyle mathbb Z the set of all integers whether positive negative or zero Z 2 1 0 1 2 3 displaystyle mathbf Z 2 1 0 1 2 3 Q displaystyle mathbf Q or Q displaystyle mathbb Q the set of all rational numbers that is the set of all proper and improper fractions Q ab a b Z b 0 displaystyle mathbf Q left frac a b mid a b in mathbf Z b neq 0 right For example 7 4 Q and 5 5 1 Q R displaystyle mathbf R or R displaystyle mathbb R the set of all real numbers including all rational numbers and all irrational numbers which include algebraic numbers such as 2 displaystyle sqrt 2 that cannot be rewritten as fractions as well as transcendental numbers such as p and e C displaystyle mathbf C or C displaystyle mathbb C the set of all complex numbers C a bi a b R for example 1 2i C Each of the above sets of numbers has an infinite number of elements Each is a subset of the sets listed below it Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs respectively For example Q displaystyle mathbf Q represents the set of positive rational numbers FunctionsA function or mapping from a set A to a set B is a rule that assigns to each input element of A an output that is an element of B more formally a function is a special kind of relation one that relates each element of A to exactly one element of B A function is called injective or one to one if it maps any two different elements of A to different elements of B surjective or onto if for every element of B there is at least one element of A that maps to it and bijective or a one to one correspondence if the function is both injective and surjective in this case each element of A is paired with a unique element of B and each element of B is paired with a unique element of A so that there are no unpaired elements An injective function is called an injection a surjective function is called a surjection and a bijective function is called a bijection or one to one correspondence CardinalityThe cardinality of a set S denoted S is the number of members of S For example if B blue white red then B 3 Repeated members in roster notation are not counted so blue white red blue white 3 too More formally two sets share the same cardinality if there exists a bijection between them The cardinality of the empty set is zero Infinite sets and infinite cardinality The list of elements of some sets is endless or infinite For example the set N displaystyle mathbb N of natural numbers is infinite In fact all the special sets of numbers mentioned in the section above are infinite Infinite sets have infinite cardinality Some infinite cardinalities are greater than others Arguably one of the most significant results from set theory is that the set of real numbers has greater cardinality than the set of natural numbers Sets with cardinality less than or equal to that of N displaystyle mathbb N are called countable sets these are either finite sets or countably infinite sets sets of the same cardinality as N displaystyle mathbb N some authors use countable to mean countably infinite Sets with cardinality strictly greater than that of N displaystyle mathbb N are called uncountable sets However it can be shown that the cardinality of a straight line i e the number of points on a line is the same as the cardinality of any segment of that line of the entire plane and indeed of any finite dimensional Euclidean space The continuum hypothesis The continuum hypothesis formulated by Georg Cantor in 1878 is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and the cardinality of a straight line In 1963 Paul Cohen proved that the continuum hypothesis is independent of the axiom system ZFC consisting of Zermelo Fraenkel set theory with the axiom of choice ZFC is the most widely studied version of axiomatic set theory Power setsThe power set of a set S is the set of all subsets of S The empty set and S itself are elements of the power set of S because these are both subsets of S For example the power set of 1 2 3 is 1 2 3 1 2 1 3 2 3 1 2 3 The power set of a set S is commonly written as P S or 2S If S has n elements then P S has 2n elements For example 1 2 3 has three elements and its power set has 23 8 elements as shown above If S is infinite whether countable or uncountable then P S is uncountable Moreover the power set is always strictly bigger than the original set in the sense that any attempt to pair up the elements of S with the elements of P S will leave some elements of P S unpaired There is never a bijection from S onto P S PartitionsA partition of a set S is a set of nonempty subsets of S such that every element x in S is in exactly one of these subsets That is the subsets are pairwise disjoint meaning any two sets of the partition contain no element in common and the union of all the subsets of the partition is S Basic operationsThe complement of A in U Suppose that a universal set U a set containing all elements being discussed has been fixed and that A is a subset of U The complement of A is the set of all elements of U that do not belong to A It may be denoted Ac or A In set builder notation Ac a U a A displaystyle A text c a in U a notin A The complement may also be called the absolute complement to distinguish it from the relative complement below Example If the universal set is taken to be the set of integers then the complement of the set of even integers is the set of odd integers The union of A and B denoted A BThe intersection of A and B denoted A BThe set difference A BThe symmetric difference of A and B Given any two sets A and B their union A B is the set of all things that are members of A or B or both their intersection A B is the set of all things that are members of both A and B If A B then A and B are said to be disjoint the set difference A B also written A B is the set of all things that belong to A but not B Especially when B is a subset of A it is also called the relative complement of B in A With Bc as the absolute complement of B in the universal set U A B A Bc their symmetric difference A D B is the set of all things that belong to A or B but not both One has ADB A B B A displaystyle A Delta B A setminus B cup B setminus A their cartesian product A B is the set of all ordered pairs a b such that a is an element of A and b is an element of B Examples 1 2 3 3 4 5 1 2 3 4 5 1 2 3 3 4 5 3 1 2 3 3 4 5 1 2 1 2 3 D 3 4 5 1 2 4 5 a b 1 2 3 a 1 a 2 a 3 b 1 b 2 b 3 The operations above satisfy many identities For example one of De Morgan s laws states that A B A B that is the elements outside the union of A and B are the elements that are outside A and outside B The cardinality of A B is the product of the cardinalities of A and B This is an elementary fact when A and B are finite When one or both are infinite multiplication of cardinal numbers is defined to make this true The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring and intersection as the multiplication of the ring ApplicationsSets are ubiquitous in modern mathematics For example structures in abstract algebra such as groups fields and rings are sets closed under one or more operations One of the main applications of naive set theory is in the construction of relations A relation from a domain A to a codomain B is a subset of the Cartesian product A B For example considering the set S rock paper scissors of shapes in the game of the same name the relation beats from S to S is the set B scissors paper paper rock rock scissors thus x beats y in the game if the pair x y is a member of B Another example is the set F of all pairs x x2 where x is real This relation is a subset of R R because the set of all squares is subset of the set of all real numbers Since for every x in R one and only one pair x is found in F it is called a function In functional notation this relation can be written as F x x2 Principle of inclusion and exclusionThe inclusion exclusion principle for two finite sets states that the size of their union is the sum of the sizes of the sets minus the size of their intersection The inclusion exclusion principle is a technique for counting the elements in a union of two finite sets in terms of the sizes of the two sets and their intersection It can be expressed symbolically as A B A B A B displaystyle A cup B A B A cap B A more general form of the principle gives the cardinality of any finite union of finite sets A1 A2 A3 An A1 A2 A3 An A1 A2 A1 A3 An 1 An 1 n 1 A1 A2 A3 An displaystyle begin aligned left A 1 cup A 2 cup A 3 cup ldots cup A n right amp left left A 1 right left A 2 right left A 3 right ldots left A n right right amp left left A 1 cap A 2 right left A 1 cap A 3 right ldots left A n 1 cap A n right right amp ldots amp left 1 right n 1 left left A 1 cap A 2 cap A 3 cap ldots cap A n right right end aligned HistoryThe concept of a set emerged in mathematics at the end of the 19th century The German word for set Menge was coined by Bernard Bolzano in his work Paradoxes of the Infinite Passage with a translation of the original set definition of Georg Cantor The German word Menge for set is translated with aggregate here Georg Cantor one of the founders of set theory gave the following definition at the beginning of his Beitrage zur Begrundung der transfiniten Mengenlehre A set is a gathering together into a whole of definite distinct objects of our perception or our thought which are called elements of the set Bertrand Russell introduced the distinction between a set and a class a set is a class but some classes such as the class of all sets are not sets see Russell s paradox When mathematicians deal with what they call a manifold aggregate Menge ensemble or some equivalent name it is common especially where the number of terms involved is finite to regard the object in question which is in fact a class as defined by the enumeration of its terms and as consisting possibly of a single term which in that case is the class Naive set theory The foremost property of a set is that it can have elements also called members Two sets are equal when they have the same elements More precisely sets A and B are equal if every element of A is an element of B and every element of B is an element of A this property is called the extensionality of sets As a consequence e g 2 4 6 and 4 6 4 2 represent the same set Unlike sets multisets can be distinguished by the number of occurrences of an element e g 2 4 6 and 4 6 4 2 represent different multisets while 2 4 6 and 6 4 2 are equal Tuples can even be distinguished by element order e g 2 4 6 and 6 4 2 represent different tuples The simple concept of a set has proved enormously useful in mathematics but paradoxes arise if no restrictions are placed on how sets can be constructed Russell s paradox shows that the set of all sets that do not contain themselves i e x x is a set and x x cannot exist Cantor s paradox shows that the set of all sets cannot exist Naive set theory defines a set as any well defined collection of distinct elements but problems arise from the vagueness of the term well defined Axiomatic set theory In subsequent efforts to resolve these paradoxes since the time of the original formulation of naive set theory the properties of sets have been defined by axioms Axiomatic set theory takes the concept of a set as a primitive notion The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular mathematical propositions statements about sets using first order logic According to Godel s incompleteness theorems however it is not possible to use first order logic to prove any such particular axiomatic set theory is free from paradox See alsoAlgebra of sets Alternative set theory Category of sets Class set theory Family of sets Fuzzy set Mereology Principia MathematicaNotesSome typographical variants are occasionally used such as ϕ or ϕ The term unit set is also occasionally used ReferencesCantor Georg Jourdain Philip E B Translator 1915 Contributions to the founding of the theory of transfinite numbers New York Dover Publications 1954 English translation By an aggregate Menge we are to understand any collection into a whole Zusammenfassung zu einem Ganzen M of definite and separate objects m of our intuition or our thought Here p 85 P K Jain Khalil Ahmad Om P Ahuja 1995 Functional Analysis New Age International p 1 ISBN 978 81 224 0801 0 Samuel Goldberg 1 January 1986 Probability An Introduction Courier Corporation p 2 ISBN 978 0 486 65252 8 Thomas H Cormen Charles E Leiserson Ronald L Rivest Clifford Stein 2001 Introduction To Algorithms MIT Press p 1070 ISBN 978 0 262 03293 3 Halmos 1960 p 1 Hilbert David 1926 Uber das Unendliche Mathematische Annalen vol 95 pp 161 190 doi 10 1007 BF01206605 JFM 51 0044 02 S2CID 121888793 Aus dem Paradies das Cantor uns geschaffen soll uns niemand vertreiben konnen Translated in Van Heijenoort Jean On the infinite Harvard University Press Maddocks J R 2004 Lerner K Lee Lerner Brenda Wilmoth eds The Gale Encyclopedia of Science Gale pp 3587 3589 ISBN 0 7876 7559 8 Devlin Keith J 1981 Sets Functions and Logic Basic concepts of university mathematics Springer pp 32 33 ISBN 978 0 412 22660 1 Halmos 1960 p 2 Marek Capinski Peter E Kopp 2004 Measure Integral and Probability Springer Science amp Business Media p 2 ISBN 978 1 85233 781 0 Set Symbols www mathsisfun com Retrieved 2020 08 19 Stoll Robert 1974 Sets Logic and Axiomatic Theories W H Freeman and Company pp 5 ISBN 9780716704577 Aggarwal M L 2021 1 Sets Understanding ISC Mathematics Class XI Vol 1 Arya Publications Avichal Publishing Company p A 3 Sourendra Nath De January 2015 Unit 1 Sets and Functions 1 Set Theory Chhaya Ganit Ekadash Shreni Scholar Books Pvt Ltd p 5 Halmos 1960 p 8 K T Leung Doris Lai chue Chen 1 July 1992 Elementary Set Theory Part I II Hong Kong University Press p 27 ISBN 978 962 209 026 2 Charles Roberts 24 June 2009 Introduction to Mathematical Proofs A Transition CRC Press p 45 ISBN 978 1 4200 6956 3 David Johnson David B Johnson Thomas A Mowry June 2004 Finite Mathematics Practical Applications Docutech Version W H Freeman p 220 ISBN 978 0 7167 6297 3 Ignacio Bello Anton Kaul Jack R Britton 29 January 2013 Topics in Contemporary Mathematics Cengage Learning p 47 ISBN 978 1 133 10742 2 Susanna S Epp 4 August 2010 Discrete Mathematics with Applications Cengage Learning p 13 ISBN 978 0 495 39132 6 A Kanamori The Empty Set the Singleton and the Ordered Pair p 278 Bulletin of Symbolic Logic vol 9 no 3 2003 Accessed 21 August 2023 Stephen B Maurer Anthony Ralston 21 January 2005 Discrete Algorithmic Mathematics CRC Press p 11 ISBN 978 1 4398 6375 6 Introduction to Sets www mathsisfun com Retrieved 2020 08 19 D Van Dalen H C Doets H De Swart 9 May 2014 Sets Naive Axiomatic and Applied A Basic Compendium with Exercises for Use in Set Theory for Non Logicians Working and Teaching Mathematicians and Students Elsevier Science p 1 ISBN 978 1 4831 5039 0 Alfred Basta Stephan DeLong Nadine Basta 1 January 2013 Mathematics for Information Technology Cengage Learning p 3 ISBN 978 1 285 60843 3 Laura Bracken Ed Miller 15 February 2013 Elementary Algebra Cengage Learning p 36 ISBN 978 0 618 95134 5 Frank Ruda 6 October 2011 Hegel s Rabble An Investigation into Hegel s Philosophy of Right Bloomsbury Publishing p 151 ISBN 978 1 4411 7413 0 John F Lucas 1990 Introduction to Abstract Mathematics Rowman amp Littlefield p 108 ISBN 978 0 912675 73 2 Weisstein Eric W Set Wolfram MathWorld Retrieved 2020 08 19 Ralph C Steinlage 1987 College Algebra West Publishing Company ISBN 978 0 314 29531 6 Felix Hausdorff 2005 Set Theory American Mathematical Soc p 30 ISBN 978 0 8218 3835 8 Peter Comninos 6 April 2010 Mathematical and Computer Programming Techniques for Computer Graphics Springer Science amp Business Media p 7 ISBN 978 1 84628 292 8 Halmos 1960 p 3 George Tourlakis 13 February 2003 Lectures in Logic and Set Theory Volume 2 Set Theory Cambridge University Press p 137 ISBN 978 1 139 43943 5 Yiannis N Moschovakis 1994 Notes on Set Theory Springer Science amp Business Media ISBN 978 3 540 94180 4 Arthur Charles Fleck 2001 Formal Models of Computation The Ultimate Limits of Computing World Scientific p 3 ISBN 978 981 02 4500 9 William Johnston 25 September 2015 The Lebesgue Integral for Undergraduates The Mathematical Association of America p 7 ISBN 978 1 939512 07 9 Karl J Smith 7 January 2008 Mathematics Its Power and Utility Cengage Learning p 401 ISBN 978 0 495 38913 2 John Stillwell 16 October 2013 The Real Numbers An Introduction to Set Theory and Analysis Springer Science amp Business Media ISBN 978 3 319 01577 4 David Tall 11 April 2006 Advanced Mathematical Thinking Springer Science amp Business Media p 211 ISBN 978 0 306 47203 9 Cantor Georg 1878 Ein Beitrag zur Mannigfaltigkeitslehre Journal fur die Reine und Angewandte Mathematik 1878 84 242 258 doi 10 1515 crll 1878 84 242 inactive 1 November 2024 a href wiki Template Cite journal title Template Cite journal cite journal a CS1 maint DOI inactive as of November 2024 link Cohen Paul J December 15 1963 The Independence of the Continuum Hypothesis Proceedings of the National Academy of Sciences of the United States of America 50 6 1143 1148 Bibcode 1963PNAS 50 1143C doi 10 1073 pnas 50 6 1143 JSTOR 71858 PMC 221287 PMID 16578557 Halmos 1960 p 19 Halmos 1960 p 20 Edward B Burger Michael Starbird 18 August 2004 The Heart of Mathematics An invitation to effective thinking Springer Science amp Business Media p 183 ISBN 978 1 931914 41 3 Toufik Mansour 27 July 2012 Combinatorics of Set Partitions CRC Press ISBN 978 1 4398 6333 6 Halmos 1960 p 28 Jose Ferreiros 16 August 2007 Labyrinth of Thought A History of Set Theory and Its Role in Modern Mathematics Birkhauser Basel ISBN 978 3 7643 8349 7 Steve Russ 9 December 2004 The Mathematical Works of Bernard Bolzano OUP Oxford ISBN 978 0 19 151370 1 William Ewald William Bragg Ewald 1996 From Kant to Hilbert Volume 1 A Source Book in the Foundations of Mathematics OUP Oxford p 249 ISBN 978 0 19 850535 8 Paul Rusnock Jan Sebestik 25 April 2019 Bernard Bolzano His Life and Work OUP Oxford p 430 ISBN 978 0 19 255683 7 Georg Cantor Nov 1895 Beitrage zur Begrundung der transfiniten Mengenlehre 1 Mathematische Annalen in German 46 4 481 512 Bertrand Russell 1903 The Principles of Mathematics chapter VI Classes Jose Ferreiros 1 November 2001 Labyrinth of Thought A History of Set Theory and Its Role in Modern Mathematics Springer Science amp Business Media ISBN 978 3 7643 5749 8 Raatikainen Panu 2022 Zalta Edward N ed Godel s Incompleteness Theorems Stanford Encyclopedia of Philosophy Metaphysics Research Lab Stanford University Retrieved 2024 06 03 ReferencesDauben Joseph W 1979 Georg Cantor His Mathematics and Philosophy of the Infinite Boston Harvard University Press ISBN 0 691 02447 2 Halmos Paul R 1960 Naive Set Theory Princeton N J Van Nostrand ISBN 0 387 90092 6 Stoll Robert R 1979 Set Theory and Logic Mineola N Y Dover Publications ISBN 0 486 63829 4 Velleman Daniel 2006 How To Prove It A Structured Approach Cambridge University Press ISBN 0 521 67599 5 External linksThe dictionary definition of set at Wiktionary Cantor s Beitrage zur Begrundung der transfiniten Mengenlehre in German Portals MathematicsArithmetic