VIDEOS 1 TO 50

Partition of a set

Published: 2009/09/15

Channel: Wei Ching Quek

Equivalence Classes Partition a Set Proof

Published: 2014/11/30

Channel: The Math Sorcerer

21 Partition of a set

Published: 2015/05/06

Channel: Sujay Krishna

(Abstract Algebra 1) Partitions and Equivalence Relations

Published: 2013/12/09

Channel: learnifyable

Find the number of ways to partition a set of 2 elements.

Published: 2013/04/20

Channel: Jeffery Dittenber

(Abstract Algebra 1) Definition of a Partition

Published: 2013/12/03

Channel: learnifyable

Set Partition Refinement Lattice

Published: 2013/10/10

Channel: wolframmathematica

Set Active / Hidden Partitions

Published: 2012/12/24

Channel: ESRepair

Partition of a set

Published: 2014/11/06

Channel: Audiopedia

Programming Interview: Balanced Partition of Array (Dynamic Programming)

Published: 2012/08/03

Channel: saurabhschool

Set Partition as primary or logical partition | MiniTool Partition Wizard Official Video Guide

Published: 2014/07/23

Channel: minitoolsolution

Bootmgr is Missing Set Active Partition

Published: 2013/09/26

Channel: TechRePair

Set Partition Statistics moment formulas and normality Part 1

Published: 2014/12/06

Channel: Experimental mathematics

Partition of a set

Published: 2016/01/22

Channel: WikiAudio

2. 3-Partition I

Published: 2015/07/14

Channel: MIT OpenCourseWare

Windows 10 - How To: Partition Hard Drives

Published: 2015/08/26

Channel: WinBeta (On MSFT)

█▶How to Fully install Kali Linux on a partition an properly set it up for ①③③⑦ use◀█

Published: 2015/07/31

Channel: imSoGettingBANNED

How to Install and Partition Windows 7

Published: 2011/01/21

Channel: DIY PC Repairs

How to set password for your each windows 7 partition

Published: 2013/01/28

Channel: Ramya K

How to Partition, Format External Hard Drive on Mac for Mac and Windows, How to Set Up Time Machine

Published: 2012/06/25

Channel: Vineet Agarwal

Set (Partition) Model

Published: 2015/10/02

Channel: Mildred Osborne

How to Make a Partition on Windows 7

Published: 2010/03/17

Channel: Allan Baguinon

Partitions (Screencast 7.3.3)

Published: 2012/11/27

Channel: GVSUmath

move app to system partition and set rw-r-r permission using root explorer app

Published: 2013/04/24

Channel: Herbert Lagonoy

Hard Disk Partitioning While Installing Windows

Published: 2013/06/13

Channel: mancisdahbasah

Partition Theorem (Total Probability Theorem) & Bayes Theorem | Example

Published: 2015/02/23

Channel: WelshBeastMaths

Apache Hive - 01 Partition an existing data set according to one or more partition keys

Published: 2016/06/30

Channel: itversity

How to set Hackintosh partition active

Published: 2013/11/09

Channel: riswan azhari

Apache Hive - 02 Partition an existing data set according to one or more partition keys

Published: 2016/06/30

Channel: itversity

How to properly configure the SSD as boot drive and HDD as storage drive

Published: 2013/05/08

Channel: NCIX Tech Tips

How To set Active Partition On a Mac/Hackintosh All Versions SL, M, ML, M

Published: 2014/04/17

Channel: MyMidnightTutorials

Programming Interviews: Partition Array in Equal Halves with Minimum Sum Difference

Published: 2012/07/01

Channel: saurabhschool

The Last Days Of The British Raj - Pakistan India Partition 1947 - by roothmens

Published: 2016/08/11

Channel: Roothmens Armageddon

How to create Partition in Windows 10 | EASY

Published: 2015/03/16

Channel: Cat and Andrew

Set Partition Label - MiniTool Partition Wizard official Video Guide

Published: 2014/07/04

Channel: minitoolsolution

Spectral Graph Partitioning Finding a Partition (Advanced) | Stanford

Published: 2016/04/13

Channel: Video Tutorials - All in One

Set Allocation Unit Size to Format Partition to FAT32 on Windows-FAT32 Format

Published: 2016/07/20

Channel: TheAppCut

Partition Based Clustering 02 - K Means Clustering Method

Published: 2015/08/26

Channel: Omar Sobh

Example of Countable Partition

Published: 2011/09/29

Channel: MathDoctorBob

Network Analysis. Lecture 9. Graph partitioning algorithms

Published: 2015/03/10

Channel: Leonid Zhukov

Pakistan And India Partition 1947 - The Day India Burned - by roothmens

Published: 2012/12/23

Channel: Roothmens Armageddon

100% Solved:Setup was Unable to Create a New System Partition[Windows 7, 8 &10]

Published: 2016/01/08

Channel: How I Solve

How To Set Up New Partition For Mac OS X Beta Without Erasing File Data

Published: 2014/06/03

Channel: HotTips!

Windows 8 Bootable Recovery Partition

Published: 2014/09/10

Channel: Britec09

Partition Set Assembly Tutorial

Published: 2016/02/23

Channel: Conductive Containers

Partition Based Clustering 04 - The K Medoids Clustering Method

Published: 2015/08/26

Channel: Omar Sobh

Beyoncé - Partition (Explicit Video)

Published: 2014/02/25

Channel: beyonceVEVO

How to Set a Primary Partition as Active Partition (No Audio)

Published: 2013/03/09

Channel: Networking Lab

Property Set Asides and Partition Actions

Published: 2015/03/31

Channel: Eric Roy

How to Create a Partition on Windows 10

Published: 2015/09/13

Channel: www.2tech.me

From Wikipedia, the free encyclopedia

For the partition calculus of sets, see infinitary combinatorics.

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.

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]}

*P*does not contain the empty set.- The union of the sets in
*P*is equal to*X*. (The sets in*P*are said to**cover***X*.) - The intersection of any two distinct sets in
*P*is empty. (We say the elements of*P*are pairwise disjoint.)

In mathematical notation, these conditions can be represented as

- if and then ,

where is the empty set.

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.

- 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:
- { {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}.

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.

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 *α* ≤ *ρ*.

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 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}.

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.

The total number of partitions of an *n*-element set is the Bell number *B _{n}*. The first several Bell numbers are

and have the exponential generating function

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 the two numbers to the left and above left of each 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 *C _{n}*, given by

Wikimedia Commons has media related to .Set partitions |

- Exact cover
- Cluster analysis
- Weak ordering (ordered set partition)
- Equivalence relation
- Partial equivalence relation
- Partition refinement
- List of partition topics
- Lamination (topology)
- Rhyme schemes by set partition

**^**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**^**Naive Set Theory (1960).*Halmos, Paul R.*Springer. p. 28. ISBN 9780387900926.**^**Lucas, John F. (1990).*Introduction to Abstract Mathematics*. Rowman & Littlefield. p. 187. ISBN 9780912675732.**^**Brualdi,*pp*. 44–45**^**Schechter,*p*. 54**^**Birkhoff, Garrett (1995),*Lattice Theory*, Colloquium Publications,**25**(3rd ed.), American Mathematical Society, p. 95, ISBN 9780821810255.

- 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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