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In [[mathematics]], a '''singleton''', also known as a '''unit set''',<ref name="Stoll">{{Cite book  | last = Stoll  | first = Robert  | authorlink =  | coauthors =  | title = Sets, Logic and Axiomatic Theories  | publisher = W. H. Freeman and Company  | series =  | volume =  | edition =  | year = 1961  | location =  | pages = 5–6  | language =  | url =  | doi =  | id =  | isbn =  | mr =  | zbl =  | jfm = }}</ref> is a [[Set (mathematics)|set]] with [[unique|exactly one]] element. For example, the set {0} is a singleton.
This Halloween, make a unique custom. Instead of it being a day, or days, of indulging in a favorite sugary treats, create it a day for making healthy resolutions.<br><br>There is the [http://safedietplans.com/bmr-calculator bmr calculator] or Weight reduction calculator. BMR or Basal Metabolic Rate is the quantity of calories that you shred while at rest or without doing anything including sitting idly inside a seat or sleeping on a bed. But BMR Calculators will likely not just calculate the idle modes however, also adds up the active methods during the day. Additionally, they usually include inside summing up the gender, age plus height of the individual. Why for this really is because those that are older more likely have lower metabolism than those that are young. This really is also true for those that are taller and have more muscle.<br><br>The basal metabolic rate is essentially the amount of calories the body must survive for one day whilst doing general bodily functions like breathing and pumping blood etc. Taking in less calories then this can force the body to burn fat as energy. There's is a calculator on the calculator page connected above.<br><br>Our bodies require 1,800 to 2,000 calories a day in order to function properly. Some individuals want less or even more, nevertheless this might be the average. So don't try to drop a noticeable amount of calories at once. There is not any should place the body from starvation. If you do, the metabolic rate actually slows down, that makes the entire task harder.<br><br>Then let's calculate the number of calories the have to support the daily escapades. Simply increase a calculated bmr by 1.2 should you don't exercise at all, 1.375 should you exercise lightly 1 to 3 occasions per week, 1.55 in the event you exercise moderately 3 to 5 occasions per week, 1.725 when you exercise hard 6 to 7 times per week, or 1.9 should you have a physically demanding job plus exercise daily or are training for a sports competition like a marathon.<br><br>To calculate a BMR, try using this free online calculator: http://www.caloriesperhour.com/index_burn.html . When you have a results, jot them down on a piece of paper for reference.<br><br>So you're suggesting which when do a metabolic fashion exercise I can eat whatever I desire? Absolutely not! The types of foods we eat are as important because the kind of workout you do. When you eat is only because important as what you eat. In the next limited blogs I is discussing foods to eat plus those to avoid, because well the proper instances to consume for the many successful fat loss. Be found on the lookout for the fresh N-Motion Fitness recipe section on nmotionfitnessllc.com.
 
The term is also used for a 1-[[tuple]] (a [[sequence]] with one element).
 
==Properties==
Within the framework of [[Zermelo–Fraenkel set theory]], the [[axiom of regularity]] guarantees that no set is an element of itself. This implies that a singleton is necessarily distinct from the element it contains,<ref name="Stoll"/> thus 1 and {1} are not the same thing, and the [[empty set]] is distinct from the set containing only the empty set. A set such as <nowiki>{{1, 2, 3}}</nowiki> is a singleton as it contains a single element (which itself is a set, however, not a singleton).
 
A set is a singleton [[if and only if]] its [[cardinality]] is {{num|1}}. In the [[Set-theoretic definition of natural numbers|standard set-theoretic construction of the natural numbers]], the number 1 is ''defined'' as the singleton {0}.
 
In [[axiomatic set theory]], the existence of singletons is a consequence of the [[axiom of pairing]]: for any set ''A'', the axiom applied to ''A'' and ''A'' asserts the existence of {''A'', ''A''}, which is the same as the singleton {''A''} (since it contains ''A'', and no other set, as an element).
 
If ''A'' is any set and ''S'' is any singleton, then there exists precisely one [[function (mathematics)|function]] from ''A'' to ''S'', the function sending every element of ''A'' to the single element of ''S''. Thus every singleton is a [[terminal object]] in the [[category of sets]].
 
== Applications ==
In [[topology]], a space is a [[T1 space]] if and only if every singleton is [[closed set|closed]].
 
Structures built on singletons often serve as [[terminal object]]s or [[zero object]]s of various [[category (category theory)|categories]]:
* The statement above shows that the singleton sets are precisely the terminal objects in the category '''[[category of sets|Set]]''' of [[set (mathematics)|set]]s. No other sets are terminal.
* Any singleton can be turned into a [[topological space]] in just one way (all subsets are open). These singleton topological spaces are terminal objects in the category of topological spaces and continuous functions. No other spaces are terminal in that category.
* Any singleton can be turned into a [[group (mathematics)|group]] in just one way (the unique element serving as [[identity element]]). These singleton groups are [[Initial object|zero object]]s in the category of groups and [[group homomorphism]]s. No other groups are terminal in that category.
 
== Definition by indicator functions ==
Let <math>S</math> be a [[Class (set theory)|class]] defined by an [[indicator function]]
:<math>b: X \to \{0, 1\}</math>.
Then <math>S</math> is called a '''''singleton''''' if and only if there is some {{math|''y'' ∈ ''X''}} such that for all {{math|''x'' ∈ ''X''}},
:<math>b(x) = (x = y) \,</math>.
 
Traditionally, this definition was introduced by [[Alfred North Whitehead|Whitehead]] and [[Bertrand Russell|Russell]]<ref>{{cite book | first=Alfred North | last=Whitehead | coauthors=Bertrand Russell | year=1910 | title=[[Principia Mathematica]] | page=37 }}</ref> along with the definition of the [[1 (number)|natural number 1]], as
:<math>1 \ \overset{\underset{\mathrm{def}}{}}{=} \ \hat{\alpha}\{(\exists x) . \alpha = \iota \jmath x\}</math>, where <math>\iota \jmath x \ \overset{\underset{\mathrm{def}}{}}{=} \ \hat{y}(y = x)</math>.
 
==See also==
*[[Class (set theory)]]
 
==References==
<div class="references"><references/></div>
 
[[Category:Basic concepts in set theory]]
[[Category:One]]

Revision as of 11:50, 18 February 2014

This Halloween, make a unique custom. Instead of it being a day, or days, of indulging in a favorite sugary treats, create it a day for making healthy resolutions.

There is the bmr calculator or Weight reduction calculator. BMR or Basal Metabolic Rate is the quantity of calories that you shred while at rest or without doing anything including sitting idly inside a seat or sleeping on a bed. But BMR Calculators will likely not just calculate the idle modes however, also adds up the active methods during the day. Additionally, they usually include inside summing up the gender, age plus height of the individual. Why for this really is because those that are older more likely have lower metabolism than those that are young. This really is also true for those that are taller and have more muscle.

The basal metabolic rate is essentially the amount of calories the body must survive for one day whilst doing general bodily functions like breathing and pumping blood etc. Taking in less calories then this can force the body to burn fat as energy. There's is a calculator on the calculator page connected above.

Our bodies require 1,800 to 2,000 calories a day in order to function properly. Some individuals want less or even more, nevertheless this might be the average. So don't try to drop a noticeable amount of calories at once. There is not any should place the body from starvation. If you do, the metabolic rate actually slows down, that makes the entire task harder.

Then let's calculate the number of calories the have to support the daily escapades. Simply increase a calculated bmr by 1.2 should you don't exercise at all, 1.375 should you exercise lightly 1 to 3 occasions per week, 1.55 in the event you exercise moderately 3 to 5 occasions per week, 1.725 when you exercise hard 6 to 7 times per week, or 1.9 should you have a physically demanding job plus exercise daily or are training for a sports competition like a marathon.

To calculate a BMR, try using this free online calculator: http://www.caloriesperhour.com/index_burn.html . When you have a results, jot them down on a piece of paper for reference.

So you're suggesting which when do a metabolic fashion exercise I can eat whatever I desire? Absolutely not! The types of foods we eat are as important because the kind of workout you do. When you eat is only because important as what you eat. In the next limited blogs I is discussing foods to eat plus those to avoid, because well the proper instances to consume for the many successful fat loss. Be found on the lookout for the fresh N-Motion Fitness recipe section on nmotionfitnessllc.com.

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