Share
VIDEOS 1 TO 50
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
Partition set | What is partition set
Partition set | What is partition set
Published: 2017/12/04
Channel: Perfect Computer Engineer
Partition of Set
Partition of Set
Published: 2017/11/03
Channel: Competition Corner
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
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
21 Partition of a set
21 Partition of a set
Published: 2015/05/06
Channel: Sujay Krishna
Equivalence Classes Partition a Set Proof
Equivalence Classes Partition a Set Proof
Published: 2014/11/30
Channel: The Math Sorcerer
Equivalence Relations And Partitions | Math | Chegg Tutors
Equivalence Relations And Partitions | Math | Chegg Tutors
Published: 2016/04/20
Channel: Chegg
Partition of a set
Partition of a set
Published: 2014/11/06
Channel: Audiopedia
Partition of a set
Partition of a set
Published: 2016/01/22
Channel: WikiAudio
8. Covering Partition - Relations - Gate
8. Covering Partition - Relations - Gate
Published: 2017/06/11
Channel: PacketPrep
Windows 10 - How To: Partition Hard Drives
Windows 10 - How To: Partition Hard Drives
Published: 2015/08/26
Channel: On MSFT
2. 3-Partition I
2. 3-Partition I
Published: 2015/07/14
Channel: MIT OpenCourseWare
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
Set Partition Model
Set Partition Model
Published: 2017/11/18
Channel: Klair Loper
Set Theory Equivalence Relations and Partitions
Set Theory Equivalence Relations and Partitions
Published: 2013/06/24
Channel: Complete GATE
How to create Partition on Windows 10 | Partition Hard Drives
How to create Partition on Windows 10 | Partition Hard Drives
Published: 2017/03/04
Channel: ProgrammingKnowledge
DISJOINT SET,PARTITION OF SET,ORDER SET,SET LAWS
DISJOINT SET,PARTITION OF SET,ORDER SET,SET LAWS
Published: 2017/10/31
Channel: University Academy- Formerly-IP University CSE/IT
Set (Partition) Model
Set (Partition) Model
Published: 2015/10/02
Channel: Mildred Osborne
Partition of Set
Partition of Set
Published: 2018/01/03
Channel: easy for you
How to Install and Partition Windows 7
How to Install and Partition Windows 7
Published: 2011/01/21
Channel: DIY PC Repairs
Bootmgr is Missing Set Active Partition
Bootmgr is Missing Set Active Partition
Published: 2013/09/26
Channel: TechRePair
Programming Interview: Balanced Partition of Array (Dynamic Programming)
Programming Interview: Balanced Partition of Array (Dynamic Programming)
Published: 2012/08/03
Channel: saurabhschool
HOW TO LOCK YOUR HARD DRIVE PARTITION WITH PASSWORD
HOW TO LOCK YOUR HARD DRIVE PARTITION WITH PASSWORD
Published: 2016/05/09
Channel: RAJ ROUSHAN
no. of ways of partition
no. of ways of partition
Published: 2017/06/06
Channel: Doubtnut
Error SET PARTITION odin al instalar Android Oreo (RESUELTO)
Error SET PARTITION odin al instalar Android Oreo (RESUELTO)
Published: 2018/02/14
Channel: Crack Heros
How to Format , Partition, Transfer files , Set up time machine - external Hard Drive on Mac
How to Format , Partition, Transfer files , Set up time machine - external Hard Drive on Mac
Published: 2017/10/27
Channel: acguevara
Example of Countable Partition
Example of Countable Partition
Published: 2011/09/30
Channel: MathDoctorBob
How to set password for your each windows 7 partition
How to set password for your each windows 7 partition
Published: 2013/01/28
Channel: Ramya K
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
Set Partition Refinement Lattice
Set Partition Refinement Lattice
Published: 2013/10/10
Channel: wolframmathematica
Division: Set Partition Model
Division: Set Partition Model
Published: 2017/11/17
Channel: Kayla Lorenz
How to delete protected partition in Windows
How to delete protected partition in Windows
Published: 2017/11/30
Channel: Helpful IT
Solved: Setup was unable to create a new system partition or locate an existing system partition
Solved: Setup was unable to create a new system partition or locate an existing system partition
Published: 2016/07/28
Channel: AvoidErrors
move app to system partition and set rw-r-r permission using root explorer app
move app to system partition and set rw-r-r permission using root explorer app
Published: 2013/04/25
Channel: Lapu Lapu
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
How to properly configure the SSD as boot drive and HDD as storage drive
How to properly configure the SSD as boot drive and HDD as storage drive
Published: 2013/05/09
Channel: NCIX Tech Tips
Set Partition Label - MiniTool Partition Wizard official Video Guide
Set Partition Label - MiniTool Partition Wizard official Video Guide
Published: 2014/07/04
Channel: minitoolsolution
How to Make a Partition on Windows 7
How to Make a Partition on Windows 7
Published: 2010/03/18
Channel: Allan Baguinon
Partition Wizard: Phân Vùng, Đổi Kí Tự Phân Vùng, Set Active/Logical/Primary
Partition Wizard: Phân Vùng, Đổi Kí Tự Phân Vùng, Set Active/Logical/Primary
Published: 2017/05/21
Channel: Quyền Tống Ngọc
Partition of a Set - bury P
Partition of a Set - bury P
Published: 2011/07/27
Channel: sbickleaf
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
Partition Set Assembly Tutorial
Partition Set Assembly Tutorial
Published: 2016/02/23
Channel: Conductive Containers
2016 07 22 07 11 map   reduce   Set   Map   Cons method   partition method   sortBy   flatMap   fora
2016 07 22 07 11 map reduce Set Map Cons method partition method sortBy flatMap fora
Published: 2016/10/16
Channel: pramod narayana
Cross Partition Calculator
Cross Partition Calculator
Published: 2011/12/18
Channel: MathCelebritydotcom
Fixed - Change Active Partition - No Operating System Error
Fixed - Change Active Partition - No Operating System Error
Published: 2015/09/08
Channel: EasyComputerUse
ES 3 Math Whole Number Partition Model as Division
ES 3 Math Whole Number Partition Model as Division
Published: 2014/09/02
Channel: TheWCPSSAcademics
Property Set Asides and Partition Actions
Property Set Asides and Partition Actions
Published: 2015/03/31
Channel: Eric Roy
NEXT
GO TO RESULTS [51 .. 100]

WIKIPEDIA ARTICLE

From Wikipedia, the free encyclopedia
Jump to: navigation, search
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.

Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called a setoid, typically in type theory and proof theory.

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 if it has the following property: If four elements a, b, c and d of N having a < b < c < d satisfy a ~ c and b ~ d, then a ~ b ~ c ~ d. The name comes from the following equivalent definition: Imagine the elements 1, 2, ..., n of N drawn as the n vertices of a regular n-gon (in counterclockwise order). A partition can then be visualized by drawing each block as a polygon (whose vertices are the elements of the block). The partition is then noncrossing if and only if these polygons do not intersect.

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. ^ Halmos, Paul (1960). Naive Set Theory R. Springer. p. 28. ISBN 9780387900926. 
  3. ^ Lucas, John F. (1990). Introduction to Abstract Mathematics. Rowman & Littlefield. p. 187. ISBN 9780912675732. 
  4. ^ Brualdi 2004, pp. 44–45.
  5. ^ Schechter 1997, 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. 

Disclaimer

None of the audio/visual content is hosted on this site. All media is embedded from other sites such as GoogleVideo, Wikipedia, YouTube etc. Therefore, this site has no control over the copyright issues of the streaming media.

All issues concerning copyright violations should be aimed at the sites hosting the material. This site does not host any of the streaming media and the owner has not uploaded any of the material to the video hosting servers. Anyone can find the same content on Google Video or YouTube by themselves.

The owner of this site cannot know which documentaries are in public domain, which has been uploaded to e.g. YouTube by the owner and which has been uploaded without permission. The copyright owner must contact the source if he wants his material off the Internet completely.

Powered by YouTube
Wikipedia content is licensed under the GFDL and (CC) license