Wikipedia

Set (mathematics)

A set of polygons in an Euler diagram

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

Sets are ubiquitous in modern mathematics. The more specialized subject of set theory is part of the foundations of mathematics.[4]

Origin [ edit ]

The concept of a set emerged in mathematics at the end of the 19th century.[6] The German word Menge, rendered as "set" in English, was coined by Bernard Bolzano in his work The 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 was one of the founders of set theory. He gave the following definition of a set 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 [Anschauung] or of our thought—which are called elements of the set.

Naïve set theory [ edit ]

The foremost basic property of a set is that it can have elements. Another essential property of sets is that two sets are equal (one set equals the other, so the two sets are in fact one and the same) if and only if every element of each set is an element of the other. This property is called the extensionality of sets.[11]

The simple concept of a set has proved enormously useful in mathematics, but it suffers from inconsistencies at the most fundamental level. A loose notion that allows any property without restriction to define a collection, leads to several paradoxes, most notably:

  • Russell's paradox – It shows that the "set of all sets that do not contain themselves," i.e. the "set" {x|x is a set and xx} does not exist.
  • Cantor's paradox – It shows that "the set of all sets" cannot exist.

Naïve set theory defines a set as any well-defined collection of distinct elements. Problems arise from the vague meaning of the term well-defined.

Axiomatic set theory [ edit ]

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 notation [ edit ]

Mathematical texts commonly use capital letters[13][4][14] in italic such as A, B, C to denote sets.[14][15]

A set can be defined either intensionally, extensionally[16] or ostensively.

Semantic definition [ edit ]

The simplest intensional method of defining a set is by using a rule or semantic description:[17][16]

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

Roster notation [ edit ]

Sets are not limited to collections of elements following simple rules, such as the sets in the examples above, however. Roster notation (or enumeration notation) is a method of defining a set by listing (or enumerating) the members of the set,[18][19][20][21][22] enclosing the list of members in curly brackets:

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

This is an example of enumerative definition.

Unlike a sequence, a tuple or a permutation of a set, the order in which the elements of a set are listed in roster notation is irrelevant, so {6, 11} is the same set as {11, 6}, and {2, 6, 4}, {4, 2, 6}, {4, 6, 2}, {6, 2, 4} or {6, 4, 2} all represent the same set.[23][15][24]

For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using ellipsis ("...").[25][26] For instance, the set of the first thousand positive integers may be specified in roster notation as:

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

where the ellipsis indicates that the list continues by following the established pattern.[25]

Infinite sets in roster notation [ edit ]

Some sets have an endless list of elements. These are called infinite sets. For example, the set of integers, including positive, negative and zero, is an infinite set. In roster notation, this set can be written with just one ellipsis:

{0, 1, -1, 2, -2, 3, -3...}

or alternatively, using two:

{...-2, -1, 0, 1, 2, 3...}

Set-builder notation [ edit ]

Set-builder notation is another intensional method of describing a set, which is often found in mathematical texts.[16] The set is specified as a selection from a larger set, determined by a condition involving the elements.[27][28] For example, a set F can be defined as follows:

F

or:

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.[29]

Membership [ edit ]

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".[30][14][31]

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 set [ edit ]

The empty set or null set, the set of no members, denoted { } or ∅, is unique.[32][14][33] Other notations are also in use (see empty set).[14]

Singleton sets [ edit ]

A set with exactly one element, x, is a unit set, or singleton, {x}.[5]

The set {x} is semantically distinct from the element x. (Halmos[34] draws the analogy that a box containing a hat is not the same as the hat.)

Subsets [ edit ]

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.[35] BA means B contains A, B includes A, or B is a superset of A; BA is equivalent to AB.[36][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.[27]

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),[37][30] while others reserve AB and BA for cases where A is a proper subset of B.[35]

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,[32] and every set is a subset of itself:[37]

  • ∅ ⊆ A.
  • AA.

Euler and Venn diagrams [ edit ]

A is a subset of B

An Euler diagram is a graphical representation of a set as a closed loop, enclosing its elements, or the relationships between different sets, as closed loops. If two sets have no members in common, the loops do not overlap.

This is distinct from a Venn diagram, which shows all possible relations between two or more sets, with each loop overlapping the others.

Special sets of numbers in mathematics [ edit ]

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.[38] These include:[14]

  • N or , denoting the set of all natural numbers: N = {0, 1, 2, 3, ...} (some authors exclude 0)[38]
  • Z or , denoting the set of all integers (whether positive, negative or zero): Z = {..., −2, −1, 0, 1, 2, ...}[38]
  • Q or , denoting 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, 1/4 ∈ Q and 11/6 ∈ Q, and since every integer n can be expressed as the fraction n/1, all integers are members of this set ()[38]
  • R or , denoting the set of all real numbers, including all rational numbers, together with all irrational numbers (that is, algebraic numbers that cannot be rewritten as fractions such as 2, as well as transcendental numbers such as π, e)[38]
  • C or , denoting the set of all complex numbers: C = {a + bi | a, bR}, for example, 1 + 2iC[38]

Each of the above sets of numbers has an infinite number of elements, and each can be considered to be a proper 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.

Mappings and one-to-one correspondence [ edit ]

In mathematics, a mapping or function from set A to a set B is a relation between the two sets, which relates each element of set A with exactly one element of set B. A one-to-one correspondence or bijection is a mapping where each element of set A is paired with exactly one element of set B, and each element of set B is paired with exactly one element of set A, so that there are no unpaired elements.

Cardinality [ edit ]

The cardinality of a set S, denoted |S|, is the number of members of S.[39] For example, if B = {blue, white, red}, then |B| = 3. Repeated members in roster notation are not counted,[40][41] 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.[42]

Infinite sets and infinite cardinality [ edit ]

The list of elements of some sets is endless, or infinite. For example, the set ℕ of natural numbers is infinite.[27] 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 ℕ 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.[43] 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.[44]

The Continuum Hypothesis [ edit ]

The Continuum Hypothesis of Georg Cantor in 1878 stated that there are no sets with cardinality between that of countable sets and that of a straight line.[45] In 1963, Paul Cohen proved the Continuum Hypothesis is independent of Zermelo–Fraenkel set theory with the axiom of choice (ZFC, the most widely-studied version of axiomatic set theory).[46]

Power sets [ edit ]

The power set of a set S is the set of all subsets of S.[27] 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.[27][47][14][15]

The power set of a finite set with n elements has 2n elements.[48] 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, 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).)[49]

Partitions [ edit ]

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.[50][51]

Basic operations [ edit ]

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

Unions [ edit ]

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.

Intersections [ edit ]

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.

Complements [ edit ]

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 product [ edit ]

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 |.

Applications [ edit ]

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 [ edit ]

The inclusion-exclusion principle can be 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 laws [ edit ]

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 also [ edit ]

Notes [ edit ]

  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 m "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. ^ 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.
  17. ^ Halmos 1960, p. 4.
  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. ^ Charles Roberts (24 June 2009). Introduction to Mathematical Proofs: A Transition. CRC Press. p. 45. ISBN 978-1-4200-6956-3.
  21. ^ Ignacio Bello; Anton Kaul; Jack R. Britton (29 January 2013). Topics in Contemporary Mathematics. Cengage Learning. p. 47. ISBN 1-133-10742-7.
  22. ^ Susanna S. Epp (4 August 2010). Discrete Mathematics with Applications. Cengage Learning. p. 13. ISBN 0-495-39132-8.
  23. ^ Stephen B. Maurer; Anthony Ralston (21 January 2005). Discrete Algorithmic Mathematics. CRC Press. p. 11. ISBN 978-1-4398-6375-6.
  24. ^ 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.
  25. ^ a b Alfred Basta; Stephan DeLong; Nadine Basta (1 January 2013). Mathematics for Information Technology. Cengage Learning. p. 3. ISBN 1-285-60843-7.
  26. ^ Laura Bracken; Ed Miller (15 February 2013). Elementary Algebra. Cengage Learning. p. 36. ISBN 0-618-95134-2.
  27. ^ a b c d e John F. Lucas (1990). Introduction to Abstract Mathematics. Rowman & Littlefield. p. 108. ISBN 978-0-912675-73-2.
  28. ^ Weisstein, Eric W. "Set". mathworld.wolfram.com. Retrieved 2020-08-19.
  29. ^ Ralph C. Steinlage (1987). College Algebra. West Publishing Company. ISBN 978-0-314-29531-6.
  30. ^ a b Marek Capinski; Peter E. Kopp (2004). Measure, Integral and Probability. Springer Science & Business Media. p. 2. ISBN 978-1-85233-781-0.
  31. ^ "Set Symbols". www.mathsisfun.com. Retrieved 2020-08-19.
  32. ^ a b Halmos 1960, p. 8.
  33. ^ 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.
  34. ^ Halmos 1960, Sect.2.
  35. ^ a b Felix Hausdorff (2005). Set Theory. American Mathematical Soc. p. 30. ISBN 978-0-8218-3835-8.
  36. ^ 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.
  37. ^ a b Halmos 1960, p. 3.
  38. ^ 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.
  39. ^ Yiannis N. Moschovakis (1994). Notes on Set Theory. Springer Science & Business Media. ISBN 978-3-540-94180-4.
  40. ^ Arthur Charles Fleck (2001). Formal Models of Computation: The Ultimate Limits of Computing. World Scientific. p. 3. ISBN 978-981-02-4500-9.
  41. ^ William Johnston (25 September 2015). The Lebesgue Integral for Undergraduates. The Mathematical Association of America. p. 7. ISBN 978-1-939512-07-9.
  42. ^ Karl J. Smith (7 January 2008). Mathematics: Its Power and Utility. Cengage Learning. p. 401. ISBN 0-495-38913-7.
  43. ^ 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.
  44. ^ David Tall (11 April 2006). Advanced Mathematical Thinking. Springer Science & Business Media. p. 211. ISBN 978-0-306-47203-9.
  45. ^ 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.
  46. ^ 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.
  47. ^ Halmos 1960, p. 19.
  48. ^ Halmos 1960, p. 20.
  49. ^ 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.
  50. ^ Toufik Mansour (27 July 2012). Combinatorics of Set Partitions. CRC Press. ISBN 978-1-4398-6333-6.
  51. ^ Halmos 1960, p. 28.

References [ edit ]

External links [ edit ]

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