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| [[File:Indiabundleware.jpg|thumb|A set of stamps partitioned into bundles: No stamp is in two bundles, and no bundle is empty.]]
| | Quick weight loss diets have been about since the world's initially dieter looked at their abdomen and thought, "I have to lose a few pounds - NOW." Even though experience has shown which fast "fad" diets usually result in temporary weight loss, dieters are nevertheless looking for the Holy Grail: A diet that lets them lose weight quick plus keep it off.<br><br>There are very a few sites which make it simple to find out how countless calories we want for fat repair plus fat loss. Just type [http://safedietplans.com/bmr-calculator bmr calculator] into a look engine plus follow the steps they give you. BMR stands for basal metabolic rate, and acquiring this amount usually aid provide we an idea of how countless calories you'll need to consume for your particular goals. You'll be asked to answer certain questions about your activity level plus maybe even the degree of fat loss we want to achieve. Some BMR calculators will go so far as to supply you with the amounts of proteins, carbohydrates and fats we should consume as well. They're a very useful tool!<br><br>Drinking water also increases a basal metabolic rate. The body has to process the water, and inside doing this burns more calories. Studies furthermore show which drinking cold water burns more calories because the body first has to bring the water up to a internal temperature.<br><br>Horsegram is powdered to a good consistency. Heat sour buttermilk plus add 100 gm of horsegram powder to this plus create a consistent paste. Apply this paste onto fat deposits found on the body and massage vigorously in upward strokes. Horsegram is recognized to lower body fat very effectively.Take hot water bathtub after half an hr. Use 'eladhi choornam' instead of soap. Add a few drops of water to this choornam plus prepare a thick paste and employ it for bathing purposes.<br><br>Folks always ask how countless calories must they be consuming in a day to keep there weight, youll just have to do a little math. The initially step is calculating your bmr that this might be the amount of stamina the body takes in plus must function properly. We use about 60% of the calories we consume everyday for the normal bodily functions such as simply by being alive and breathing the others that influence the BMR are height, weight, age and also sex.<br><br>Psychologists say which folks that experience a sense of insecurity, which can be due to real or imagined threats, might find solace and security in food. This results inside food cravings plus emotional binge eating.<br><br>Nutritionists employ the first 2 formulas as a guideline to compute simple calorie demands. They never compute extra calories utilized in exercise. The last one computes the calories different weight need based on activity level. |
| [[File:Set partitions 5; circles.svg|thumb|The [[Bell number|52]] partitions of a set with 5 elements]]
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| [[File:Genji chapter symbols groupings of 5 elements.svg|thumb|The traditional Japanese symbols for the chapters of the ''[[Tale of Genji]]'' are based on the 52 ways of partitioning five elements.]]
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| In [[mathematics]], a '''partition''' of a [[Set (mathematics)|set]] ''X'' is a division of ''X'' as a union of non-overlapping and non-empty subsets, sometimes called "'''parts'''" or "'''blocks'''" or "'''cells'''". More formally, these "cells" are both [[collectively exhaustive]] and [[mutually exclusive]] with respect to the set being partitioned.
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| == Definition ==
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| A partition of a set ''X'' is a set of [[nonempty]] [[subset]]s of ''X'' such that every element ''x'' in ''X'' is in exactly one of these subsets<ref>{{Cite book|author=Naive Set Theory|title=Halmos, Paul R.|publisher=Springer|year=1960|isbn=9780387900926|page=28|url=http://books.google.com/books?id=x6cZBQ9qtgoC&pg=PA28}}</ref> (i.e., ''X'' is a [[disjoint union]] of the subsets). | |
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| Equivalently, a [[family of sets]] ''P'' is a partition of ''X'' if and only if all of the following conditions hold:<ref>{{cite book|author=Lucas, John F.|title=Introduction to Abstract Mathematics|publisher=Rowman & Littlefield|year=1990|isbn=9780912675732|page=187|url=http://books.google.com/books?id=jklsb5JUgoQC&pg=PA187}}</ref>
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| #''P'' does not contain the empty set.
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| # The [[union (set theory)|union]] of the sets in ''P'' is equal to ''X''. (The sets in ''P'' are said to '''cover''' ''X''.)
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| # The [[intersection (set theory)|intersection]] of any two distinct sets in ''P'' is empty. (We say the elements of ''P'' are [[pairwise disjoint]].)
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| In mathematical notation, these conditions can be represented as
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| # <math>\varnothing \notin P</math>
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| # <math>\bigcup_{A\in P} A = X</math>
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| # if <math>A,B \in P</math> and <math>A\neq B</math> then <math>A \cap B = \varnothing</math>,
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| where <math>\varnothing</math> is the [[empty set]].
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| The sets in ''P'' are called the blocks, parts or cells of the partition.<ref name="brualdi44_45">Brualdi, ''pp''. 44–45</ref>
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| The '''rank''' of ''P'' is |''X''| − |''P''|, if ''X'' is finite.
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| == Examples ==
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| *Every [[singleton set]] {''x''} has exactly one partition, namely { {''x''} }.
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| *For any nonempty set ''X'', ''P'' = {''X''} is a partition of ''X'', called the '''trivial partition'''.
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| *For any non-empty [[proper subset]] ''A'' of a set ''U'', the set ''A'' together with its [[complement (set theory)|complement]] form a partition of ''U'', namely, {''A'', ''U''−''A''}.
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| *The set { 1, 2, 3 } has these five partitions:
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| ** { {1}, {2}, {3} }, sometimes written 1|2|3.
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| ** { {1, 2}, {3} }, or 12|3.
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| ** { {1, 3}, {2} }, or 13|2.
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| ** { {1}, {2, 3} }, or 1|23.
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| ** { {1, 2, 3} }, or 123 (in contexts where there will be no confusion with the number).
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| *The following are not partitions of { 1, 2, 3 }:
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| ** { {}, {1, 3}, {2} } is not a partition (of any set) because one of its elements is the [[empty set]].
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| ** { {1, 2}, {2, 3} } is not a partition (of any set) because the element 2 is contained in more than one block.
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| ** { {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}.
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| == Partitions and equivalence relations ==
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| For any [[equivalence relation]] on a set ''X'', the set of its [[equivalence class]]es is a partition of ''X''. Conversely, from any partition ''P'' of ''X'', we can define an equivalence relation on ''X'' by setting {{nowrap|''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.<ref name="schechter54">Schechter, ''p''. 54</ref>
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| 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.
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| == Refinement of partitions ==
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| [[File:Set partitions 4; Hasse; circles.svg|thumb|left|300px|Partitions of a 4-set ordered by [[Partition refinement|refinement]]]]
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| 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''.
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| This ''finer-than'' relation on the set of partitions of ''X'' is a [[partially ordered set|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 (order)|lattice]], and more specifically (for partitions of a finite set) it is a [[geometric lattice]].<ref>{{citation|title=Lattice Theory|volume=25|series=Colloquium Publications|publisher=American Mathematical Society|first=Garrett|last=Birkhoff|authorlink=Garrett Birkhoff|edition=3rd|year=1995|isbn=9780821810255|page=95|url=http://books.google.com/books?id=0Y8d-MdtVwkC&pg=PA95}}.</ref> The ''partition lattice'' of a 4-element set has 15 elements and is depicted in the [[Hasse diagram]] on the left.
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| Based on the [[cryptomorphism]] between geometric lattices and [[matroid]]s, this lattice of partitions of a finite set corresponds to a matroid in which the base set of the matroid consists of the [[Atom (order theory)|atoms]] of the lattice, the partitions with <math>n-2</math> singleton sets and one two-element set. These atomic partitions correspond one-for-one with the edges of a [[complete graph]]. The [[Matroid#Closure_operators|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 [[Vertex (graph theory)|vertices]] of the complete graph into the [[Connected component (graph theory)|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.
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| 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 ~<sub>C</sub> – has two equivalence classes: the sets {red cards} and {black cards}. The 2-part partition corresponding to ~<sub>C</sub> has a refinement that yields the ''same-suit-as'' relation ~<sub>S</sub>, which has the four equivalence classes {spades}, {diamonds}, {hearts}, and {clubs}.
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| ==Noncrossing partitions==
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| A partition of the set ''N'' = {1, 2, ..., ''n''} with corresponding equivalence relation ~ is '''[[noncrossing partition|noncrossing]]''' 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.
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| 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.
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| == Counting partitions ==
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| The total number of partitions of an ''n''-element set is the [[Bell numbers|Bell number]] ''B<sub>n</sub>''. The first several Bell numbers are ''B''<sub>0</sub> = 1,
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| ''B''<sub>1</sub> = 1, ''B''<sub>2</sub> = 2, ''B''<sub>3</sub> = 5, ''B''<sub>4</sub> = 15, ''B''<sub>5</sub> = 52, and ''B''<sub>6</sub> = 203. Bell numbers satisfy the [[recursion]] <math>B_{n+1}=\sum_{k=0}^n {n\choose k}B_k</math>
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| and have the [[generating function|exponential generating function]]
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| :<math>\sum_{n=0}^\infty\frac{B_n}{n!}z^n=e^{e^z-1}.</math>
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| [[Image:BellNumberAnimated.gif|right|thumb|Construction of the [[Bell triangle]]]]
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| The Bell numbers may also be computed using the [[Bell triangle]]
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| 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 (mathematics)|singleton]].
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| 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''). | |
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| The number of [[noncrossing partition]]s of an ''n''-element set is the [[Catalan number]] ''C<sub>n</sub>'', given by
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| :<math>C_n={1 \over n+1}{2n \choose n}.</math>
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| ==See also==
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| {{Commons category|Set partitions}}
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| * [[Data clustering]]
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| * [[Exact cover]]
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| * [[Exponential formula]]
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| * [[Faà di Bruno formula]]
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| * [[Lamination (topology)|Lamination]]
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| * [[List of partition topics]]
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| * [[Partial equivalence relation]]
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| * [[Partition refinement]]
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| * [[Refinement (sigma algebra)]]
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| * [[Weak ordering]] (ordered set partition)
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{cite book |last= Brualdi |first= Richard A. |title= Introductory Combinatorics |edition= 4th edition |year= 2004 |publisher= Pearson Prentice Hall |isbn= 0-13-100119-1}}
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| *{{cite book |last= Schechter |first= Eric |title= Handbook of Analysis and Its Foundations |year= 1997 |publisher= Academic Press |isbn= 0-12-622760-8}}
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| [[Category:Basic concepts in set theory]]
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| [[Category:Combinatorics]]
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| [[Category:Set families]]
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Quick weight loss diets have been about since the world's initially dieter looked at their abdomen and thought, "I have to lose a few pounds - NOW." Even though experience has shown which fast "fad" diets usually result in temporary weight loss, dieters are nevertheless looking for the Holy Grail: A diet that lets them lose weight quick plus keep it off.
There are very a few sites which make it simple to find out how countless calories we want for fat repair plus fat loss. Just type bmr calculator into a look engine plus follow the steps they give you. BMR stands for basal metabolic rate, and acquiring this amount usually aid provide we an idea of how countless calories you'll need to consume for your particular goals. You'll be asked to answer certain questions about your activity level plus maybe even the degree of fat loss we want to achieve. Some BMR calculators will go so far as to supply you with the amounts of proteins, carbohydrates and fats we should consume as well. They're a very useful tool!
Drinking water also increases a basal metabolic rate. The body has to process the water, and inside doing this burns more calories. Studies furthermore show which drinking cold water burns more calories because the body first has to bring the water up to a internal temperature.
Horsegram is powdered to a good consistency. Heat sour buttermilk plus add 100 gm of horsegram powder to this plus create a consistent paste. Apply this paste onto fat deposits found on the body and massage vigorously in upward strokes. Horsegram is recognized to lower body fat very effectively.Take hot water bathtub after half an hr. Use 'eladhi choornam' instead of soap. Add a few drops of water to this choornam plus prepare a thick paste and employ it for bathing purposes.
Folks always ask how countless calories must they be consuming in a day to keep there weight, youll just have to do a little math. The initially step is calculating your bmr that this might be the amount of stamina the body takes in plus must function properly. We use about 60% of the calories we consume everyday for the normal bodily functions such as simply by being alive and breathing the others that influence the BMR are height, weight, age and also sex.
Psychologists say which folks that experience a sense of insecurity, which can be due to real or imagined threats, might find solace and security in food. This results inside food cravings plus emotional binge eating.
Nutritionists employ the first 2 formulas as a guideline to compute simple calorie demands. They never compute extra calories utilized in exercise. The last one computes the calories different weight need based on activity level.