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In mathematics, a partition of a set X is a division of X into non-overlapping "parts" or "blocks" or "cells" that cover all of X. More formally, these "cells" are both collectively exhaustive and mutually exclusive with respect to the set being partitioned. -from the creater, wshun 02:20, 24 Dec 2004 (UTC) File links The following pages link to this file: Partition of a set Categories: FAL images ...
An Euler diagram does not need to show all possible intersections. ...
Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...
In probability theory, a set of events is collectively exhaustive if at least one of the events must occur. ...
In logic, two mutually exclusive (or mutual exclusive according to some sources) propositions are propositions that logically cannot both be true. ...
Definition 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. In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set. ...
A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ...
Equivalently, a set P of subsets of X, is a partition of X if
- No element of P is empty. (NB - some definitions do not include this)
- The union of the elements of P is equal to X. (We say the elements of P cover X.)
- The intersection of any two elements of P is empty. (We say the elements of P are pairwise disjoint.)
The elements of P are sometimes called the blocks or parts of the partition.[1] The empty set is the set containing no elements. ...
In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...
In mathematics, two sets are said to be disjoint if they have no element in common. ...
Examples - Every singleton set {x} has exactly one partition, namely { {x} }.
- For any nonempty set X, P = {X} is a partition of X.
- The empty set has exactly one partition, namely one with no blocks.
- For any non-empty proper subset A of a set U, this A together with its complement is a partition of U.
- If we do not use axiom 1, then the above example generalizes so that any subset (empty or not) together with its complement is a partition.
- The set { 1, 2, 3 } has these five partitions.
- { {1}, {2}, {3} }, sometimes denoted by 1/2/3.
- { {1, 2}, {3} }, sometimes denoted by 12/3.
- { {1, 3}, {2} }, sometimes denoted by 13/2.
- { {1}, {2, 3} }, sometimes denoted by 1/23.
- { {1, 2, 3} }, sometimes denoted by 123.
- Note that
- { {}, {1,3}, {2} } is not a partition if we are using axiom 1 (because it contains the empty set); otherwise it is a partition of {1, 2, 3}.
- { {1,2}, {2, 3} } is not a partition (of any set) because the element 2 is contained in more than one distinct subset.
- { {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}.
The empty set is the set containing no elements. ...
A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y...
In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...
In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...
Partitions and equivalence relations If an equivalence relation is given on the set X, then the set of all equivalence classes forms a partition of X. Conversely, if a partition P is given on X, we can define an equivalence relation on X by writing x ~ y if there exists a member of P which contains both x and y. The notions of "equivalence relation" and "partition" are thus essentially equivalent.[2] In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ...
In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...
Look up if in Wiktionary, the free dictionary. ...
Partial ordering of the lattice of partitions Given two partitions π and ρ of a given set X, we say that π is finer than ρ, or, equivalently, that ρ is coarser than π, if π splits the set X into smaller blocks than ρ does, i.e. if every element of π is a subset of some element of ρ. In that case, one writes π ≤ ρ.
The relation of "being-finer-than" is a partial order on the set of all partitions of the set X, and indeed even a complete lattice. In case n = 4, the partial order of the set of all 15 partitions is depicted in this Hasse diagram: In mathematics, especially order theory, a partially ordered set (or poset) is a set equipped with a partial order relation. ...
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). ...
In the mathematical discipline known as order theory, a Hasse diagram (pronounced HAHS uh, named after Helmut Hasse (1898–1979)) is a simple picture of a finite partially ordered set, forming a drawing of the transitive reduction of the partial order. ...
Image File history File links This is a lossless scalable vector image. ...
Noncrossing partitions 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. In combinatorial mathematics, the topic of noncrossing partitions has assumed some importance because of (among other things) its application to the theory free probability. ...
Free probability is a mathematical theory which studies non-commutative random variables. ...
The number of partitions The Bell number Bn, named in honor of Eric Temple Bell, is the number of different partitions of a set with n elements. The first several Bell numbers are B0 = 1, B1 = 1, B2 = 2, B3 = 5, B4 = 15, B5 = 52, B6 = 203. The Bell numbers, named in honor of Eric Temple Bell, are a sequence of integers arising in combinatorics that begins thus (sequence A000110 in OEIS): In general, Bn is the number of partitions of a set of size n. ...
Eric Temple Bell (1883 - 1960) was a mathematician born in Scotland who lived in the USA from 1903 until his death. ...
The exponential generating function for Bell numbers is In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ...
 Bell numbers satisfy the recursion  . A visual form of recursion known as the Droste effect. ...
The Stirling number of the second kind S(n, k) is the number of partitions of a set of size n into k blocks. In mathematics, Stirling numbers of the second kind, together with Stirling numbers of the first kind, are one of the two types of Stirling numbers. ...
The number of partitions of a set of size n corresponding to the integer partition In mathematics, a partition of a positive integer n is a way of writing n as a sum of positive integers. ...
 of n is the Faà di Bruno coefficient // The formula Faà di Brunos formula is an identity in mathematics generalizing the chain rule to higher derivatives, named in honor of Francesco Faà di Bruno (1825–1888), who was (in chronological order) a military officer, a mathematician, and a priest, and was beatified by the Pope a century...
 The number of noncrossing partitions of a set of size n is the nth Catalan number, given by In combinatorial mathematics, the topic of noncrossing partitions has assumed some importance because of (among other things) its application to the theory free probability. ...
In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. ...
See also Clustering is the classification of objects into different groups, or more precisely, the partitioning of a data set into subsets (clusters), so that the data in each subset (ideally) share some common trait - often proximity according to some defined distance measure. ...
In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ...
In combinatorial mathematics, the exponential formula states that for any formal power series of the form we have , where , and the index π runs through the list of all partitions { B1, ..., Bk } of the set { 1, ..., n }. For example, , because there is one partition of the set { 1, 2, 3 } that...
This is a list of partition topics, in the mathematical sense. ...
In mathematics, a partial equivalence relation (often abbreviated as PER) R on a set X is a relation which is symmetric and transitive. ...
Notes - ^ Brualdi, pp. 44-45
- ^ Schechter, p. 54
References - Brualdi, Richard A. (2004). Introductory Combinatorics, 4th edition, Pearson Prentice Hall. ISBN 0131001191.
- Schechter, Eric (1997). Handbook of Analysis and Its Foundations. Academic Press. ISBN 0126227608.
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