Set (mathematics)

In mathematics, a set is a collection of distinct elements.[1][2][3] The elements that make up a set can be any kind of things: people, letters of the alphabet, numbers, points in space, lines, other geometrical shapes, variables, or even other sets.[4] Two sets are equal if and only if they have precisely the same elements.[5]

A set of polygons in an Euler diagram

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.[4]

OriginEdit

The concept of a set emerged in mathematics at the end of the 19th century.[6] The German word for set, Menge, was coined by Bernard Bolzano in his work Paradoxes of the Infinite.[7][8][9]

 
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 Beiträge zur Begründung der transfiniten Mengenlehre:[10]

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.

Naïve set theoryEdit

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 a member of B, and every element of B is an element of A; this property is called the extensionality of sets.[11]

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 xx}, 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 theoryEdit

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.[12] 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.[citation needed]

How sets are defined and set notationEdit

Mathematical texts commonly denote sets by capital letters[13][4][14] in italic, such as A, B, C.[14][15] A set may also be called a collection or family, especially when its elements are themselves sets.

Semantic definitionEdit

One way to define a set is to use a rule to determine what the elements are:

Let A be the set whose members are the first four positive integers.
Let B be the set of colors of the French flag.

Such a definition is also called a semantic description.[16][17]

Roster notationEdit

Roster or enumeration notation defines a set by listing its elements between curly brackets, separated by commas:[18][19][20][21]

A = {4, 2, 1, 3}
B = {blue, white, red}.

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, in a sequence, a tuple, or a permutation of a set, the ordering of the terms matters). For example, {2, 4, 6} and {4, 6, 2} represent the same set.[22][15][23]

For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis "...".[24][25] 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 notationEdit

An infinite set is a set with an endless list of elements. To describe an infinite set in roster notation, an ellipsis is 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 notationEdit

Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements.[17][26][27] For example, a set F can be defined as follows:

F  

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.[28]

Classifying methods of definitionEdit

Philosophy uses specific terms to classify types of definitions:

  • An intensional definition uses a rule to determine membership. Semantic definitions and definitions using set-builder notation are examples.
  • An extensional definition describes a set by listing all its elements.[17] Such definitions are also called enumerative.
  • An ostensive definition is one that describes a set by giving examples of elements; a roster involving an ellipsis would be an example.

MembershipEdit

If B is a set and x is an element of B, this is written in shorthand as xB, which can also be read as "x belongs to B", or "x is in B".[11] The statement "y is not an element of B" is written as yB, which can also be read as or "y is not in B".[29][14][30]

For example, with respect to the sets A = {1, 2, 3, 4}, B = {blue, white, red}, and F = {n | n is an integer, and 0 ≤ n ≤ 19},

4 ∈ A and 12 ∈ F; and
20 ∉ F and green ∉ B.

The empty setEdit

The empty set (or null set) is the unique set that has no members. It is denoted ∅ or   or { }.[31][14][32]

Singleton setsEdit

A singleton set is a set with exactly one element; such a set may also be called a unit set.[5] Any such set can be written as {x}, where x is the element. The set {x} and the element x mean different things; Halmos[33] draws the analogy that a box containing a hat is not the same as the hat.

SubsetsEdit

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 AB.[34] BA means B contains A, B includes A, or B is a superset of A; BA is equivalent to AB.[35][14] The relationship between sets established by ⊆ is called inclusion or containment. Two sets are equal if they contain each other: AB and BA is equivalent to A = B.[26]

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 AB. Likewise, BA 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 AB and BA to mean A is any subset of B (and not necessarily a proper subset),[36][29] while others reserve AB and BA for cases where A is a proper subset of B.[34]

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,[31] and every set is a subset of itself:[36]

  • ∅ ⊆ A.
  • AA.

Euler and Venn diagramsEdit

 
A is a subset of B

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 mathematicsEdit

 
The natural numbers ℕ are contained in the integers ℤ, which are contained in the rational numbers ℚ, which are contained in the real numbers ℝ, which are contained in the complex numbers

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) or blackboard bold (e.g.  ) typeface.[37] These include[14]

  • N or  , the set of all natural numbers: N = {0, 1, 2, 3, ...} (often, authors exclude 0);[37]
  • Z or  , the set of all integers (whether positive, negative or zero): Z = {..., −2, −1, 0, 1, 2, ...};[37]
  • Q or  , the set of all rational numbers (that is, the set of all proper and improper fractions): Q = {a/b | a, bZ, b ≠ 0}. For example, -7/4 ∈ Q and 5 = 5/1 ∈ Q;[37]
  • R or  , the set of all real numbers, including all rational numbers and all irrational numbers (which include algebraic numbers such as 2 that cannot be rewritten as fractions, as well as transcendental numbers such as π and e);[37]
  • C or  , the set of all complex numbers: C = {a + bi | a, bR}, for example, 1 + 2iC.[37]

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.

FunctionsEdit

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.

CardinalityEdit

The cardinality of a set S, denoted |S|, is the number of members of S.[38] For example, if B = {blue, white, red}, then |B| = 3. Repeated members in roster notation are not counted,[39][40] so |{blue, white, red, blue, white}| = 3, too.

More formally, two sets share the same cardinality if there exists a one-to-one correspondence between them.

The cardinality of the empty set is zero.[41]

Infinite sets and infinite cardinalityEdit

The list of elements of some sets is endless, or infinite. For example, the set   of natural numbers is infinite.[26] 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. Sets with the same cardinality as   are called countable sets. 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.[42] Sets with cardinality greater than the set of natural numbers 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.[43]

The Continuum HypothesisEdit

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.[44] 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.[45] (ZFC is the most widely-studied version of axiomatic set theory.)

Power setsEdit

The power set of a set S is the set of all subsets of S.[26] 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 2P.[26][46][14][15]

The power set of a finite set with n elements has 2n elements.[47] For example, the set {1, 2, 3} contains three elements, and the power set shown above contains 23 = 8 elements.

The power set of an infinite (either countable or uncountable) set is always uncountable. Moreover, within the most widely-used frameworks of set theory, the power set of a set is always strictly "bigger" than the original set, in the sense that there is no way to pair every element of S with exactly one element of P(S). (There is never an onto map or surjection from S onto P(S).)[48]

PartitionsEdit

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.[49][50]

Basic operationsEdit

There are several fundamental operations for constructing new sets from given sets.

UnionsEdit

 
The union of A and B, denoted AB

Two sets can be "added" together. The union of A and B, denoted by A ∪ B,[14] is the set of all things that are members of either A or B.

Examples:

  • {1, 2} ∪ {1, 2} = {1, 2}.
  • {1, 2} ∪ {2, 3} = {1, 2, 3}.
  • {1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}

Some basic properties of unions:

  • AB = BA.
  • A ∪ (BC) = (AB) ∪ C.
  • A ⊆ (AB).
  • AA = A.
  • A ∪ ∅ = A.
  • AB if and only if AB = B.

IntersectionsEdit

A new set can also be constructed by determining which members two sets have "in common". The intersection of A and B, denoted by AB,[14] is the set of all things that are members of both A and B. If AB = ∅, then A and B are said to be disjoint.

 
The intersection of A and B, denoted AB.

Examples:

  • {1, 2} ∩ {1, 2} = {1, 2}.
  • {1, 2} ∩ {2, 3} = {2}.
  • {1, 2} ∩ {3, 4} = ∅.

Some basic properties of intersections:

  • AB = BA.
  • A ∩ (BC) = (AB) ∩ C.
  • ABA.
  • AA = A.
  • A ∩ ∅ = ∅.
  • AB if and only if AB = A.

ComplementsEdit

 
The relative complement
of B in A
 
The complement of A in U
 
The symmetric difference of A and B

Two sets can also be "subtracted". The relative complement of B in A (also called the set-theoretic difference of A and B), denoted by A \ B (or AB),[14] is the set of all elements that are members of A, but not members of B. It is valid to "subtract" members of a set that are not in the set, such as removing the element green from the set {1, 2, 3}; doing so will not affect the elements in the set.

In certain settings, all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \ A is called the absolute complement or simply complement of A, and is denoted by A′ or Ac.[14]

  • A′ = U \ A

Examples:

  • {1, 2} \ {1, 2} = ∅.
  • {1, 2, 3, 4} \ {1, 3} = {2, 4}.
  • If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then U \ E = E′ = O.

Some basic properties of complements include the following:

  • A \ BB \ A for AB.
  • AA′ = U.
  • AA′ = ∅.
  • (A′)′ = A.
  • ∅ \ A = ∅.
  • A \ ∅ = A.
  • A \ A = ∅.
  • A \ U = ∅.
  • A \ A′ = A and A′ \ A = A′.
  • U′ = ∅ and ∅′ = U.
  • A \ B = AB.
  • if AB then A \ B = ∅.

An extension of the complement is the symmetric difference, defined for sets A, B as

 

For example, the symmetric difference of {7, 8, 9, 10} and {9, 10, 11, 12} is the set {7, 8, 11, 12}. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring (with the empty set as neutral element) and intersection as the multiplication of the ring.

Cartesian productEdit

A new set can be constructed by associating every element of one set with every element of another set. The Cartesian product of two sets A and B, denoted by A × B,[14] is the set of all ordered pairs (a, b) such that a is a member of A and b is a member of B.

Examples:

  • {1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green)}.
  • {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.
  • {a, b, c} × {d, e, f} = {(a, d), (a, e), (a, f), (b, d), (b, e), (b, f), (c, d), (c, e), (c, f)}.

Some basic properties of Cartesian products:

  • A × = ∅.
  • A × (BC) = (A × B) ∪ (A × C).
  • (AB) × C = (A × C) ∪ (B × C).

Let A and B be finite sets; then the cardinality of the Cartesian product is the product of the cardinalities:

  • | A × B | = | B × A | = | A | × | B |.

ApplicationsEdit

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 exclusionEdit

 
The inclusion-exclusion principle is used to calculate the size of the union of sets: the size of the union is the size of the two sets, minus the size of their intersection.

The inclusion–exclusion principle is a counting technique that can be used to count the number of elements in a union of two sets—if the size of each set and the size of their intersection are known. It can be expressed symbolically as

 

A more general form of the principle can be used to find the cardinality of any finite union of sets:

 

De Morgan's lawsEdit

Augustus De Morgan stated two laws about sets.

If A and B are any two sets then,

  • (A ∪ B)′ = A′ ∩ B′

The complement of A union B equals the complement of A intersected with the complement of B.

  • (A ∩ B)′ = A′ ∪ B′

The complement of A intersected with B is equal to the complement of A union to the complement of B.

See alsoEdit

NotesEdit

  1. ^ P. K. Jain; Khalil Ahmad; Om P. Ahuja (1995). Functional Analysis. New Age International. p. 1. ISBN 978-81-224-0801-0.
  2. ^ Samuel Goldberg (1 January 1986). Probability: An Introduction. Courier Corporation. p. 2. ISBN 978-0-486-65252-8.
  3. ^ Thomas H.. Cormen; 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.
  4. ^ a b c Halmos 1960, p. 1.
  5. ^ a b Stoll, Robert (1974). Sets, Logic and Axiomatic Theories. W. H. Freeman and Company. pp. 5.
  6. ^ 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.
  7. ^ Steve Russ (9 December 2004). The Mathematical Works of Bernard Bolzano. OUP Oxford. ISBN 978-0-19-151370-1.
  8. ^ 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.
  9. ^ Paul Rusnock; Jan Sebestík (25 April 2019). Bernard Bolzano: His Life and Work. OUP Oxford. p. 430. ISBN 978-0-19-255683-7.
  10. ^ "Eine Menge, ist die Zusammenfassung bestimmter, wohlunterschiedener Objekte unserer Anschauung oder unseres Denkens – welche Elemente der Menge genannt werden – zu einem Ganzen." "Archived copy". Archived from the original on 2011-06-10. Retrieved 2011-04-22.CS1 maint: archived copy as title (link)
  11. ^ a b Halmos 1960, p. 2.
  12. ^ 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.
  13. ^ Seymor Lipschutz; Marc Lipson (22 June 1997). Schaum's Outline of Discrete Mathematics. McGraw Hill Professional. p. 1. ISBN 978-0-07-136841-4.
  14. ^ a b c d e f g h i j k l "Comprehensive List of Set Theory Symbols". Math Vault. 2020-04-11. Retrieved 2020-08-19.
  15. ^ a b c "Introduction to Sets". www.mathsisfun.com. Retrieved 2020-08-19.
  16. ^ Halmos 1960, p. 4.
  17. ^ a b c 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.
  18. ^ Charles Roberts (24 June 2009). Introduction to Mathematical Proofs: A Transition. CRC Press. p. 45. ISBN 978-1-4200-6956-3.
  19. ^ 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.
  20. ^ Ignacio Bello; Anton Kaul; Jack R. Britton (29 January 2013). Topics in Contemporary Mathematics. Cengage Learning. p. 47. ISBN 1-133-10742-7.
  21. ^ Susanna S. Epp (4 August 2010). Discrete Mathematics with Applications. Cengage Learning. p. 13. ISBN 0-495-39132-8.
  22. ^ Stephen B. Maurer; Anthony Ralston (21 January 2005). Discrete Algorithmic Mathematics. CRC Press. p. 11. ISBN 978-1-4398-6375-6.
  23. ^ 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.
  24. ^ Alfred Basta; Stephan DeLong; Nadine Basta (1 January 2013). Mathematics for Information Technology. Cengage Learning. p. 3. ISBN 1-285-60843-7.
  25. ^ Laura Bracken; Ed Miller (15 February 2013). Elementary Algebra. Cengage Learning. p. 36. ISBN 0-618-95134-2.
  26. ^ a b c d e John F. Lucas (1990). Introduction to Abstract Mathematics. Rowman & Littlefield. p. 108. ISBN 978-0-912675-73-2.
  27. ^ Weisstein, Eric W. "Set". mathworld.wolfram.com. Retrieved 2020-08-19.
  28. ^ Ralph C. Steinlage (1987). College Algebra. West Publishing Company. ISBN 978-0-314-29531-6.
  29. ^ a b Marek Capinski; Peter E. Kopp (2004). Measure, Integral and Probability. Springer Science & Business Media. p. 2. ISBN 978-1-85233-781-0.
  30. ^ "Set Symbols". www.mathsisfun.com. Retrieved 2020-08-19.
  31. ^ a b Halmos 1960, p. 8.
  32. ^ 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.
  33. ^ Halmos 1960, Sect.2.
  34. ^ a b Felix Hausdorff (2005). Set Theory. American Mathematical Soc. p. 30. ISBN 978-0-8218-3835-8.
  35. ^ 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.
  36. ^ a b Halmos 1960, p. 3.
  37. ^ a b c d e f 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.
  38. ^ Yiannis N. Moschovakis (1994). Notes on Set Theory. Springer Science & Business Media. ISBN 978-3-540-94180-4.
  39. ^ Arthur Charles Fleck (2001). Formal Models of Computation: The Ultimate Limits of Computing. World Scientific. p. 3. ISBN 978-981-02-4500-9.
  40. ^ William Johnston (25 September 2015). The Lebesgue Integral for Undergraduates. The Mathematical Association of America. p. 7. ISBN 978-1-939512-07-9.
  41. ^ Karl J. Smith (7 January 2008). Mathematics: Its Power and Utility. Cengage Learning. p. 401. ISBN 0-495-38913-7.
  42. ^ 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.
  43. ^ David Tall (11 April 2006). Advanced Mathematical Thinking. Springer Science & Business Media. p. 211. ISBN 978-0-306-47203-9.
  44. ^ Cantor, Georg (1878). "Ein Beitrag zur Mannigfaltigkeitslehre". Journal für die Reine und Angewandte Mathematik. 84 (84): 242–258. doi:10.1515/crll.1878.84.242.
  45. ^ 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.
  46. ^ Halmos 1960, p. 19.
  47. ^ Halmos 1960, p. 20.
  48. ^ 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.
  49. ^ Toufik Mansour (27 July 2012). Combinatorics of Set Partitions. CRC Press. ISBN 978-1-4398-6333-6.
  50. ^ Halmos 1960, p. 28.

ReferencesEdit

External linksEdit