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1.5: Partitions of Sets
1.5: Partitions of Sets
Published: 2013/01/25
Channel: wishizuk
(Abstract Algebra 1) Definition of a Partition
(Abstract Algebra 1) Definition of a Partition
Published: 2013/12/03
Channel: learnifyable
Equivalence Relations & Set Partitions, Part One
Equivalence Relations & Set Partitions, Part One
Published: 2015/09/17
Channel: CU Calculus
Partition of  a set
Partition of a set
Published: 2009/09/15
Channel: Wei Ching Quek
Equivalence Relations And Partitions | Math | Chegg Tutors
Equivalence Relations And Partitions | Math | Chegg Tutors
Published: 2016/04/20
Channel: Chegg
Equivalence Classes Partition a Set Proof
Equivalence Classes Partition a Set Proof
Published: 2014/11/30
Channel: The Math Sorcerer
[Mathematical Linguistics] Equivalence Relations and Partitions
[Mathematical Linguistics] Equivalence Relations and Partitions
Published: 2016/05/16
Channel: TheTrevTutor
2. 3-Partition I
2. 3-Partition I
Published: 2015/07/14
Channel: MIT OpenCourseWare
Partitions
Partitions
Published: 2015/06/09
Channel: Sara Jensen
(Abstract Algebra 1) Partitions and Equivalence Relations
(Abstract Algebra 1) Partitions and Equivalence Relations
Published: 2013/12/09
Channel: learnifyable
Set Theory Equivalence Relations and Partitions
Set Theory Equivalence Relations and Partitions
Published: 2013/06/24
Channel: Complete GATE
Balanced Partition, by Brian Dean
Balanced Partition, by Brian Dean
Published: 2017/05/30
Channel: Max Levy
set theory lesson 6 partition of a class for bsc part 1
set theory lesson 6 partition of a class for bsc part 1
Published: 2017/03/20
Channel: ocean of gyan
Find the number of ways to partition a set of 2 elements.
Find the number of ways to partition a set of 2 elements.
Published: 2013/04/21
Channel: Jeffery Dittenber
Windows 10 - How To: Partition Hard Drives
Windows 10 - How To: Partition Hard Drives
Published: 2015/08/26
Channel: On MSFT
21 Partition of a set
21 Partition of a set
Published: 2015/05/06
Channel: Sujay Krishna
Programming Interview: Balanced Partition of Array (Dynamic Programming)
Programming Interview: Balanced Partition of Array (Dynamic Programming)
Published: 2012/08/03
Channel: saurabhschool
Counting Elements, Product Sets, Partitions
Counting Elements, Product Sets, Partitions
Published: 2012/05/25
Channel: David Waldo
Partition of a set
Partition of a set
Published: 2014/11/06
Channel: Audiopedia
Apache Hive - 01 Partition an existing data set according to one or more partition keys
Apache Hive - 01 Partition an existing data set according to one or more partition keys
Published: 2016/06/30
Channel: itversity
1.5. Set Partitions
1.5. Set Partitions
Published: 2014/06/04
Channel: UCLan Cyprus | Mathematics
Partition of a set
Partition of a set
Published: 2016/01/22
Channel: WikiAudio
Subset Sum Problem Dynamic Programming
Subset Sum Problem Dynamic Programming
Published: 2015/03/29
Channel: Tushar Roy - Coding Made Simple
Equivalence Relations & Set Partitions, Part Two
Equivalence Relations & Set Partitions, Part Two
Published: 2015/09/17
Channel: CU Calculus
The Graph Partitioning Problem
The Graph Partitioning Problem
Published: 2016/06/06
Channel: Udacity
Set Partitioning in Hierarchical Trees (SPIHT) simple example part1
Set Partitioning in Hierarchical Trees (SPIHT) simple example part1
Published: 2017/05/24
Channel: help you
8. Covering Partition - Relations - Gate
8. Covering Partition - Relations - Gate
Published: 2017/06/11
Channel: Gate CS Prep
Set Partition of Raid
Set Partition of Raid
Published: 2013/11/21
Channel: Romdon Sutthikarn
Set Partition Statistics moment formulas and normality Part 1
Set Partition Statistics moment formulas and normality Part 1
Published: 2014/12/06
Channel: Experimental mathematics
Chapter 1 #2 - Sets, Partitions, and Tree Diagrams
Chapter 1 #2 - Sets, Partitions, and Tree Diagrams
Published: 2011/01/20
Channel: FiniteHelp
Partition Theorem (Total Probability Theorem) & Bayes Theorem | Example
Partition Theorem (Total Probability Theorem) & Bayes Theorem | Example
Published: 2015/02/24
Channel: WelshBeastMaths
Set Partition as primary or logical partition | MiniTool Partition Wizard Official Video Guide
Set Partition as primary or logical partition | MiniTool Partition Wizard Official Video Guide
Published: 2014/07/23
Channel: minitoolsolution
Partition Set Assembly Tutorial
Partition Set Assembly Tutorial
Published: 2016/02/24
Channel: Conductive Containers
Set (Partition) Model
Set (Partition) Model
Published: 2015/10/02
Channel: Mildred Osborne
Example of Countable Partition
Example of Countable Partition
Published: 2011/09/30
Channel: MathDoctorBob
Equivalence Relations, Equivalence Classes and Partitions
Equivalence Relations, Equivalence Classes and Partitions
Published: 2016/12/14
Channel: Iqbal Shahid
Programming Interviews: Partition Array in Equal Halves with Minimum Sum Difference
Programming Interviews: Partition Array in Equal Halves with Minimum Sum Difference
Published: 2012/07/01
Channel: saurabhschool
Unions and Partitions - Foundations of Pure Mathematics - Dr Joel Feinstein
Unions and Partitions - Foundations of Pure Mathematics - Dr Joel Feinstein
Published: 2015/02/03
Channel: University of Nottingham
Partitioning data into training and validation datasets using R
Partitioning data into training and validation datasets using R
Published: 2015/04/18
Channel: Bharatendra Rai
Small Set Partitions
Small Set Partitions
Published: 2009/07/13
Channel: wolframmathematica
Dynamic Programming - Minimum Sum Of Maximums Of Partitions
Dynamic Programming - Minimum Sum Of Maximums Of Partitions
Published: 2014/02/01
Channel: obinnaokechukwu
Partitioning Sets By Number Tool
Partitioning Sets By Number Tool
Published: 2013/04/05
Channel: EDUGains Ontario
How to Install and Partition Windows 7
How to Install and Partition Windows 7
Published: 2011/01/21
Channel: DIY PC Repairs
no. of ways of partition
no. of ways of partition
Published: 2017/06/06
Channel: Doubtnut
Partitioning of a set of cupcakes
Partitioning of a set of cupcakes
Published: 2015/04/05
Channel: Carlos F Gonzalez
Venn Diagrams and Partitions - Sets, Partitions, and Tree Diagrams - Finite Math
Venn Diagrams and Partitions - Sets, Partitions, and Tree Diagrams - Finite Math
Published: 2011/10/14
Channel: FiniteHelp
Set   Partition
Set Partition
Published: 2017/08/23
Channel: MissyMya Entertainment
Set Partition Statistics moment formulas and normality Part 2
Set Partition Statistics moment formulas and normality Part 2
Published: 2014/12/06
Channel: Experimental mathematics
Analysis 1.3 Equivalence Relations, Partitions, Definition and Proof
Analysis 1.3 Equivalence Relations, Partitions, Definition and Proof
Published: 2016/02/12
Channel: K M
How to set Hackintosh partition active
How to set Hackintosh partition active
Published: 2013/11/09
Channel: riswan azhari
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WIKIPEDIA ARTICLE

From Wikipedia, the free encyclopedia
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A set of stamps partitioned into bundles: No stamp is in two bundles, no bundle is empty, and every stamp is in a bundle.
The 52 partitions of a set with 5 elements. A colored region indicates a subset of X, forming a member of the enclosing partition. Uncolored dots indicate single-element subsets. The first shown partition contains five single-element subsets; the last partition contains one subset having five elements.
The traditional Japanese symbols for the chapters of the Tale of Genji are based on the 52 ways of partitioning five elements.[1]

In mathematics, a partition of a set is a grouping of the set's elements into non-empty subsets, in such a way that every element is included in one and only one of the subsets.

Definition[edit]

A partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly one of these subsets[2] (i.e., X is a disjoint union of the subsets).

Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold:[3]

  • The family P does not contain the empty set (that is ).
  • The union of the sets in P is equal to X (that is ). The sets in P are said to cover X.
  • The intersection of any two distinct sets in P is empty (that is ). The elements of P are said to be pairwise disjoint.

The sets in P are called the blocks, parts or cells of the partition.[4]

The rank of P is |X| − |P|, if X is finite.

Examples[edit]

  • Every singleton set {x} has exactly one partition, namely { {x} }.
  • The empty set has exactly one partition, namely .
  • For any nonempty set X, P = {X} is a partition of X, called the trivial partition.
  • For any non-empty proper subset A of a set U, the set A together with its complement form a partition of U, namely, {A, U \ A}.
  • The set { 1, 2, 3 } has these five partitions (one partition per item):
    • { {1}, {2}, {3} }, sometimes written 1|2|3.
    • { {1, 2}, {3} }, or 12|3.
    • { {1, 3}, {2} }, or 13|2.
    • { {1}, {2, 3} }, or 1|23.
    • { {1, 2, 3} }, or 123 (in contexts where there will be no confusion with the number).
  • The following are not partitions of { 1, 2, 3 }:
    • { {}, {1, 3}, {2} } is not a partition (of any set) because one of its elements is the empty set.
    • { {1, 2}, {2, 3} } is not a partition (of any set) because the element 2 is contained in more than one block.
    • { {1}, {2} } is not a partition of {1, 2, 3} because none of its blocks contains 3; however, it is a partition of {1, 2}.

Partitions and equivalence relations[edit]

For any equivalence relation on a set X, the set of its equivalence classes is a partition of X. Conversely, from any partition P of X, we can define an equivalence relation on X by setting x ~ y precisely when x and y are in the same part in P. Thus the notions of equivalence relation and partition are essentially equivalent.[5]

The axiom of choice guarantees for any partition of a set X the existence of a subset of X containing exactly one element from each part of the partition. This implies that given an equivalence relation on a set one can select a canonical representative element from every equivalence class.

Refinement of partitions[edit]

Partitions of a 4-set ordered by refinement

A partition α of a set X is a refinement of a partition ρ of X—and we say that α is finer than ρ and that ρ is coarser than α—if every element of α is a subset of some element of ρ. Informally, this means that α is a further fragmentation of ρ. In that case, it is written that αρ.

This finer-than relation on the set of partitions of X is a partial order (so the notation "≤" is appropriate). Each set of elements has a least upper bound and a greatest lower bound, so that it forms a lattice, and more specifically (for partitions of a finite set) it is a geometric lattice.[6] The partition lattice of a 4-element set has 15 elements and is depicted in the Hasse diagram on the left.

Based on the cryptomorphism between geometric lattices and matroids, this lattice of partitions of a finite set corresponds to a matroid in which the base set of the matroid consists of the atoms of the lattice, namely, the partitions with singleton sets and one two-element set. These atomic partitions correspond one-for-one with the edges of a complete graph. The matroid closure of a set of atomic partitions is the finest common coarsening of them all; in graph-theoretic terms, it is the partition of the vertices of the complete graph into the connected components of the subgraph formed by the given set of edges. In this way, the lattice of partitions corresponds to the lattice of flats of the graphic matroid of the complete graph.

Another example illustrates the refining of partitions from the perspective of equivalence relations. If D is the set of cards in a standard 52-card deck, the same-color-as relation on D – which can be denoted ~C – has two equivalence classes: the sets {red cards} and {black cards}. The 2-part partition corresponding to ~C has a refinement that yields the same-suit-as relation ~S, which has the four equivalence classes {spades}, {diamonds}, {hearts}, and {clubs}.

Noncrossing partitions[edit]

A partition of the set N = {1, 2, ..., n} with corresponding equivalence relation ~ is noncrossing[clarification needed] provided that for any two 'cells' C1 and C2, either all the elements in C1 are < than all the elements in C2 or they are all > than all the elements in C2. In other words: given distinct numbers a, b, c in N, with a < b < c, if a ~ c (they both are in a cell called C), it follows that also a ~ b and b ~ c, that is b is also in C. The lattice of noncrossing partitions of a finite set has recently taken on importance because of its role in free probability theory. These form a subset of the lattice of all partitions, but not a sublattice, since the join operations of the two lattices do not agree.

Counting partitions[edit]

The total number of partitions of an n-element set is the Bell number Bn. The first several Bell numbers are B0 = 1, B1 = 1, B2 = 2, B3 = 5, B4 = 15, B5 = 52, and B6 = 203 (sequence A000110 in the OEIS). Bell numbers satisfy the recursion

and have the exponential generating function

Construction of the Bell triangle

The Bell numbers may also be computed using the Bell triangle in which the first value in each row is copied from the end of the previous row, and subsequent values are computed by adding two numbers, the number to the left and the number to the above left of the position. The Bell numbers are repeated along both sides of this triangle. The numbers within the triangle count partitions in which a given element is the largest singleton.

The number of partitions of an n-element set into exactly k nonempty parts is the Stirling number of the second kind S(n, k).

The number of noncrossing partitions of an n-element set is the Catalan number Cn, given by

See also[edit]

Notes[edit]

  1. ^ Knuth, Donald E. (2013), "Two thousand years of combinatorics", in Wilson, Robin; Watkins, John J., Combinatorics: Ancient and Modern, Oxford University Press, pp. 7–37 
  2. ^ Naive Set Theory (1960). Halmos, Paul R. Springer. p. 28. ISBN 9780387900926. 
  3. ^ Lucas, John F. (1990). Introduction to Abstract Mathematics. Rowman & Littlefield. p. 187. ISBN 9780912675732. 
  4. ^ Brualdi, pp. 44–45
  5. ^ Schechter, p. 54
  6. ^ Birkhoff, Garrett (1995), Lattice Theory, Colloquium Publications, 25 (3rd ed.), American Mathematical Society, p. 95, ISBN 9780821810255 .

References[edit]

  • Brualdi, Richard A. (2004). Introductory Combinatorics (4th ed.). Pearson Prentice Hall. ISBN 0-13-100119-1. 
  • Schechter, Eric (1997). Handbook of Analysis and Its Foundations. Academic Press. ISBN 0-12-622760-8. 

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