Total internal reflection - Revision history

en>Ulflund: Undid revision 640956588 by 182.66.49.221 (talk)

2015-01-04T22:20:51Z

Undid revision 640956588 by 182.66.49.221 (talk)

← Older revision		Revision as of 00:20, 5 January 2015
Line 1:		Line 1:
	~~The title~~ of the ~~author~~ is ~~Garland~~. ~~Years in~~ the ~~past we moved~~ to ~~Arizona but my spouse desires us~~ to ~~move~~. ~~The preferred pastime~~ for ~~my kids~~ and me is ~~playing crochet and now I'm attempting to make cash with it~~. ~~I am~~ a ~~cashier and I'll~~ be ~~promoted soon~~.<br><br>~~Here is my website:~~ [http://~~Www~~.~~Deloro2004~~.~~com~~/do-it-~~yourself-auto-repair-suggestions/ Www~~.~~Deloro2004~~.~~com]~~

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en>Jim1138: Reverted edits by 92.98.71.76 (talk) to last revision by Jim1138 (HG)

2014-03-01T11:34:46Z

Reverted edits by 92.98.71.76 (talk) to last revision by Jim1138 (HG)

← Older revision		Revision as of 13:34, 1 March 2014
Line 1:		Line 1:
	~~'''Transfinite induction''' is an extension of [[mathematical induction]] to [[well-order\|well-ordered sets]], for example to sets~~ of ~~[[ordinal number]]s or [[cardinal number]]s.~~		The title of the author is Garland. Years in the past we moved to Arizona but my spouse desires us to move. The preferred pastime for my kids and me is playing crochet and now I'm attempting to make cash with it. I am a cashier and I'll be promoted soon.<br><br>Here is my website: [http://Www.Deloro2004.com/do-it-yourself-auto-repair-suggestions/ Www.Deloro2004.com]

	~~== Transfinite induction ==~~

	~~Let P(α) be a [[Property (philosophy)\|property]] defined for all ordinals α. Suppose that whenever P(β) is true for all β < α, then P(α) is also true (including~~ the ~~case that P(0) is true given the [[vacuously true]] statement that P(α) is true for all <math>\alpha\in\emptyset</math>). Then transfinite induction tells us that P is true for all ordinals.~~

	~~That is, if P(α) is true whenever P(β) is true for all β < α, then P(α)~~ is ~~true for all α~~. ~~Or, more practically:~~ in ~~order to prove a property P for all ordinals α, one can assume that it is already known for all smaller β < α.~~

	~~Usually~~ the ~~proof is broken down into three cases:~~

	* '''Zero case:''' Prove that <math>P(0)</math> is true.

	* '''Successor case:''' Prove that for any [[successor ordinal]] α+1, P(α+1) follows from P(α) (and, if necessary, P(β) for all β < α).

	* '''Limit case:''' Prove that for any [[limit ordinal]] λ, P(λ) follows from [P(β) for all β < λ].

	~~Notice that all three cases are identical except for the type of ordinal considered. They do not formally need~~ to ~~be considered separately,~~ but ~~in practice the proofs are typically so different as~~ to ~~require separate presentations~~. ~~Zero is sometimes considered a [[limit ordinal]]~~ and ~~then may sometimes be treated in proofs in the same case as limit ordinals.~~

	~~==Transfinite recursion==~~
	~~{{technical\|section\|date=October 2013}}~~
	~~'''Transfinite recursion'''~~ is ~~a method of constructing or defining something~~ and ~~is closely related to the concept of transfinite induction. As an example, a sequence of sets ''A'~~'~~<sub>α</sub> is defined for every ordinal α, by specifying how~~ to ~~determine ''A''<sub>α</sub> from the sequence of ''A''<sub>β</sub> for β < α~~.

	~~More formally, we can state the Transfinite Recursion Theorem as follows. Given~~ a ~~class function ''G'': ''V'' → ''V'', there exists a unique [[transfinite sequence]] ''F'': Ord → ''V'' (where Ord is the class of all ordinals) such that~~
	~~:''F''(α) = ''G''(''F'' <math>\upharpoonright</math> α) for all ordinals α.~~
	~~As in the case of induction, we may treat different types of ordinals separately: another formulation of transfinite recursion is that given a set ''g''<sub>1</sub>,~~ and ~~class functions~~ '~~'G''<sub>2</sub>, ''G''<sub>3</sub>, there exists a unique function ''F'': Ord → ''V'' such that~~
	* ''F''(0) = ''g''<sub>1</sub>,
	* ''F''(α + 1) = ''G''<sub>2</sub>(''F''(α)), for all α ∈ Ord,
	* ''F''(λ) = ''G''<sub>3</sub>(''F'' <math>\upharpoonright</math> λ), for all limit λ ≠ 0.

	~~Note that we require the domains of ''G''~~<~~sub~~>2<~~/sub>, ''G''<sub>3</sub~~> ~~to be broad enough to make the above properties meaningful. The uniqueness of the sequence satisfying these properties can be proven using transfinite induction.~~

	~~More generally, one can define objects by transfinite recursion on any [[well-founded relation]] ''R''. (''R'' need not even be a set; it can be a [[proper class]], provided it~~ is a [~~[binary relation#Relations over a set\|set-like]] relation; that is, for any ''x'', the collection of all ''y'' such that ''y R x'' must be a set.)~~

	~~==Relationship to the axiom of choice==~~
	Proofs or constructions using induction and recursion often use the [[axiom of choice]] to produce a well-ordered relation that can be treated by transfinite induction. However, if the relation in question is already well-ordered, one can often use transfinite induction without invoking the axiom of choice.<ref>In fact, the domain of the relation does not even need to be a set. It can be a proper class, provided that the relation ''R'' is set-like: ~~for any ''x'', the collection of all ''y'' such that ''y'' ''R'' ''x'' must be a set.<~~/ref> For example, many results about [[Borel sets]] are proved by transfinite induction on the ordinal rank of the set; these ranks are already well-ordered, so the axiom of choice is not needed to well-order them.

	~~The following construction of the [[Vitali set]] shows one way that the axiom of choice can be used in a proof by transfinite induction:~~
	~~: First, [[well-order]] the [[real number]]s (this is where the axiom of choice enters via the [[well-ordering theorem]]), giving a sequence <math> \langle r_{\alpha} \| \alpha < \beta \rangle <~~/~~math>, where β is an ordinal with the [[cardinality of the continuum]]~~. ~~Let ''v''<sub>0</sub> equal ''r''<sub>0</sub>~~. ~~Then let ''v''<sub>1<~~/sub> equal ''r''<sub>α<sub>1</sub></sub>, where α<sub>1</sub> is least such that ''r''<sub>α<sub>1</sub></sub> − ''v''<sub>0</sub> is not a [[rational number]]. Continue; at each step use the least real from the ''r'' sequence that does not have a rational difference with any element thus far constructed in the ''v'' sequence. Continue until all the reals in the ''r'' sequence are exhausted. The final ''v'' sequence will enumerate the Vitali set.
	~~The above argument uses the axiom of choice in an essential way at the very beginning, in order to well~~-~~order the reals. After that step, the axiom of choice is not used again.~~

	Other uses of the axiom of choice are more subtle. For example, a construction by transfinite recursion frequently will not specify a ''unique'' value for ''A''<sub>α+1</sub>, given the sequence up to α, but will specify only a ''condition'' that ''A''<sub>α+1</sub> must satisfy, and argue that there is at least one set satisfying this condition. If it is not possible to define a unique example of such a set at each stage, then it may be necessary to invoke (some form of) the axiom of choice to select one such at each step. For inductions and recursions of [[countable set\|countable]] length, the weaker [[axiom of dependent choice]] is sufficient. Because there are models of [[Zermelo–Fraenkel set theory]] of interest to set theorists that satisfy the axiom of dependent choice but not the full axiom of choice, the knowledge that a particular proof only requires dependent choice can be useful.

	~~==See also==~~
	*[[epsilon-~~induction\|∈-induction]]~~

	~~== Notes ==~~
	~~<references/>~~

	~~== References ==~~
	*{{Citation\|last=Suppes\|first=Patrick\|authorlink=Patrick Suppes\|year=1972\|title=Axiomatic set theory\|publisher=[[Dover Publications]]\|isbn=0-~~486~~-~~61630~~-~~4\|chapter=Section 7~~.~~1}}~~

	~~== External links ==~~
	*{{MathWorld \|title=Transfinite Induction \|id=TransfiniteInduction \|author=[[Jonathan Emerson\|Emerson, Jonathan]], [[Mark Lezama\|Lezama, Mark]] and [[Eric W. ~~Weisstein\|Weisstein, Eric W.]] }}~~

	~~{{Set theory}}~~

	~~[[Category:Ordinal numbers]]~~
	~~[[Category:Recursion]]~~
	~~[[Category:Mathematical induction]~~]

en>Ulflund: Undid revision 593687018 by 50.184.8.151 (talk)

2014-02-03T13:02:36Z

Undid revision 593687018 by 50.184.8.151 (talk)

← Older revision		Revision as of 15:02, 3 February 2014
Line 1:		Line 1:
	~~Your baby will expend more time within nursery than anywhere else~~, ~~so it'~~s ~~only practical you simply give him just~~ the ~~best there~~ is ~~simply~~. ~~However~~, ~~that doesn't mean that you have to break the bank~~ in order for ~~the right pieces~~ for the ~~nursery.~~<br><br>~~This baby bathtub folds away~~ for ~~straightforward storage or perhaps~~ for ~~travel~~. It is ~~larger than most bathtubs when preserving~~ the ~~earth~~ . ~~opened. The bathtub~~'~~s inclined position~~ to ~~create your baby feel attach. You can also lay~~ the ~~bottom flat for that baby to play~~. ~~This bathtub comes with~~ an ~~easy release drain plug. The bathtub~~'~~s inflatable vinyl liner will design baby feel comfortable.~~<br><br>~~Wedding planners help build a system simple~~ to ~~grasp~~ . ~~you organize budget and start~~ the ~~best prices at~~ the ~~minimum costs~~. ~~Talk honestly about budgets. Talk about what you~~'~~ll be able to produce after which you can talk to oldsters~~ and ~~other generous loved-ones who wish to contribute. Discuss who are assigned pay out~~ for ~~for what.~~<br><br><br><br>~~More than anything else~~, ~~it crucial to have your baby's safety to mind when buying nursery furniture sets~~. ~~Guaranteed~~ that the ~~nursery furniture sets match the modern safety standards in your.~~<br><br>~~A bassinet and a little~~ one ~~cradle are small scale cribs that are easily carried~~ by a ~~vehicle to~~ any ~~room~~ the ~~for her convenience~~. ~~Babies also love~~ the ~~rocking~~ and ~~swaying features within~~ the ~~bassinets and baby cradles~~. ~~These mini cribs are really simple to clean, fix and bring around with you~~, ~~even~~ if ~~outdoor drives. That~~ is the ~~reason why most parents opt to obtain mini cribs~~.<~~br><br~~>It ~~would seem~~ that ~~Weiner has~~ the ~~formula~~ for ~~scoring an effective dramatic series down apply~~. ~~First~~, ~~grab a big pot throughout~~ the ~~kitchen~~. ~~Throw into it~~ a ~~flawed yet compelling human . Mix in pressure~~, ~~conflict~~, ~~temptation~~, ~~and also~~ ~~[http:~~//s.~~zuly.de~~/~~6u4 multisitters mumbai] the world around them. Give a boil~~, ~~then simmer.~~<br><br>~~If your furry friend~~ is ~~still~~ a ~~puppy, you must buy pet beds~~ that ~~are waterproof~~. ~~Luxury waterproof pet beds~~ are ~~lined with luxurious water-resistant fabrics and trimmed with fur~~. ~~In which means you don~~'~~t want~~ to ~~worry if your puppy wets~~ the ~~dog bed~~. ~~Also~~, ~~waterproof beds~~ are ~~less prone to damages unlike those~~ will be not ~~waterproof~~, ~~which means more value~~ to ~~the.~~<br><br>~~As I lugged the extremely heavy bed~~ to ~~the parking lot~~, ~~I severely considered how We spent~~ the ~~best part~~ of ~~my infancy rollicking around on the cold~~, ~~hard floor maybe~~ the ~~warm embrace~~ of ~~my mother~~. ~~And I proved well since~~ of ~~course. But whats up! Just because However get appreciate~~ the ~~luxury~~ of a ~~new baby bedding is no reason for me to deprive my child! My baby will sleep in his~~/~~her own baby crib bedding~~.		'''Transfinite induction''' is an extension of [[mathematical induction]] to [[well-order\|well-ordered sets]], for example to sets of [[ordinal number]]s or [[cardinal number]]s.

			== Transfinite induction ==

			Let P(α) be a [[Property (philosophy)\|property]] defined for all ordinals α. Suppose that whenever P(β) is true for all β < α, then P(α) is also true (including the case that P(0) is true given the [[vacuously true]] statement that P(α) is true for all <math>\alpha\in\emptyset</math>). Then transfinite induction tells us that P is true for all ordinals.

			That is, if P(α) is true whenever P(β) is true for all β < α, then P(α) is true for all α. Or, more practically: in order to prove a property P for all ordinals α, one can assume that it is already known for all smaller β < α.

			Usually the proof is broken down into three cases:

			* '''Zero case:''' Prove that <math>P(0)</math> is true.

			* '''Successor case:''' Prove that for any [[successor ordinal]] α+1, P(α+1) follows from P(α) (and, if necessary, P(β) for all β < α).

			* '''Limit case:''' Prove that for any [[limit ordinal]] λ, P(λ) follows from [P(β) for all β < λ].

			Notice that all three cases are identical except for the type of ordinal considered. They do not formally need to be considered separately, but in practice the proofs are typically so different as to require separate presentations. Zero is sometimes considered a [[limit ordinal]] and then may sometimes be treated in proofs in the same case as limit ordinals.

			==Transfinite recursion==
			{{technical\|section\|date=October 2013}}
			'''Transfinite recursion''' is a method of constructing or defining something and is closely related to the concept of transfinite induction. As an example, a sequence of sets ''A''<sub>α</sub> is defined for every ordinal α, by specifying how to determine ''A''<sub>α</sub> from the sequence of ''A''<sub>β</sub> for β < α.

			More formally, we can state the Transfinite Recursion Theorem as follows. Given a class function ''G'': ''V'' → ''V'', there exists a unique [[transfinite sequence]] ''F'': Ord → ''V'' (where Ord is the class of all ordinals) such that
			:''F''(α) = ''G''(''F'' <math>\upharpoonright</math> α) for all ordinals α.
			As in the case of induction, we may treat different types of ordinals separately: another formulation of transfinite recursion is that given a set ''g''<sub>1</sub>, and class functions ''G''<sub>2</sub>, ''G''<sub>3</sub>, there exists a unique function ''F'': Ord → ''V'' such that
			* ''F''(0) = ''g''<sub>1</sub>,
			* ''F''(α + 1) = ''G''<sub>2</sub>(''F''(α)), for all α ∈ Ord,
			* ''F''(λ) = ''G''<sub>3</sub>(''F'' <math>\upharpoonright</math> λ), for all limit λ ≠ 0.

			Note that we require the domains of ''G''<sub>2</sub>, ''G''<sub>3</sub> to be broad enough to make the above properties meaningful. The uniqueness of the sequence satisfying these properties can be proven using transfinite induction.

			More generally, one can define objects by transfinite recursion on any [[well-founded relation]] ''R''. (''R'' need not even be a set; it can be a [[proper class]], provided it is a [[binary relation#Relations over a set\|set-like]] relation; that is, for any ''x'', the collection of all ''y'' such that ''y R x'' must be a set.)

			==Relationship to the axiom of choice==
			Proofs or constructions using induction and recursion often use the [[axiom of choice]] to produce a well-ordered relation that can be treated by transfinite induction. However, if the relation in question is already well-ordered, one can often use transfinite induction without invoking the axiom of choice.<ref>In fact, the domain of the relation does not even need to be a set. It can be a proper class, provided that the relation ''R'' is set-like: for any ''x'', the collection of all ''y'' such that ''y'' ''R'' ''x'' must be a set.</ref> For example, many results about [[Borel sets]] are proved by transfinite induction on the ordinal rank of the set; these ranks are already well-ordered, so the axiom of choice is not needed to well-order them.

			The following construction of the [[Vitali set]] shows one way that the axiom of choice can be used in a proof by transfinite induction:
			: First, [[well-order]] the [[real number]]s (this is where the axiom of choice enters via the [[well-ordering theorem]]), giving a sequence <math> \langle r_{\alpha} \| \alpha < \beta \rangle </math>, where β is an ordinal with the [[cardinality of the continuum]]. Let ''v''<sub>0</sub> equal ''r''<sub>0</sub>. Then let ''v''<sub>1</sub> equal ''r''<sub>α<sub>1</sub></sub>, where α<sub>1</sub> is least such that ''r''<sub>α<sub>1</sub></sub> − ''v''<sub>0</sub> is not a [[rational number]]. Continue; at each step use the least real from the ''r'' sequence that does not have a rational difference with any element thus far constructed in the ''v'' sequence. Continue until all the reals in the ''r'' sequence are exhausted. The final ''v'' sequence will enumerate the Vitali set.
			The above argument uses the axiom of choice in an essential way at the very beginning, in order to well-order the reals. After that step, the axiom of choice is not used again.

			Other uses of the axiom of choice are more subtle. For example, a construction by transfinite recursion frequently will not specify a ''unique'' value for ''A''<sub>α+1</sub>, given the sequence up to α, but will specify only a ''condition'' that ''A''<sub>α+1</sub> must satisfy, and argue that there is at least one set satisfying this condition. If it is not possible to define a unique example of such a set at each stage, then it may be necessary to invoke (some form of) the axiom of choice to select one such at each step. For inductions and recursions of [[countable set\|countable]] length, the weaker [[axiom of dependent choice]] is sufficient. Because there are models of [[Zermelo–Fraenkel set theory]] of interest to set theorists that satisfy the axiom of dependent choice but not the full axiom of choice, the knowledge that a particular proof only requires dependent choice can be useful.

			==See also==
			*[[epsilon-induction\|∈-induction]]

			== Notes ==
			<references/>

			== References ==
			*{{Citation\|last=Suppes\|first=Patrick\|authorlink=Patrick Suppes\|year=1972\|title=Axiomatic set theory\|publisher=[[Dover Publications]]\|isbn=0-486-61630-4\|chapter=Section 7.1}}

			== External links ==
			*{{MathWorld \|title=Transfinite Induction \|id=TransfiniteInduction \|author=[[Jonathan Emerson\|Emerson, Jonathan]], [[Mark Lezama\|Lezama, Mark]] and [[Eric W. Weisstein\|Weisstein, Eric W.]] }}

			{{Set theory}}

			[[Category:Ordinal numbers]]
			[[Category:Recursion]]
			[[Category:Mathematical induction]]

202.142.105.228 at 19:14, 26 July 2012

2012-07-26T19:14:22Z

New page

Your baby will expend more time within nursery than anywhere else, so it's only practical you simply give him just the best there is simply. However, that doesn't mean that you have to break the bank in order for the right pieces for the nursery. This baby bathtub folds away for straightforward storage or perhaps for travel. It is larger than most bathtubs when preserving the earth . opened. The bathtub's inclined position to create your baby feel attach. You can also lay the bottom flat for that baby to play. This bathtub comes with an easy release drain plug. The bathtub's inflatable vinyl liner will design baby feel comfortable. Wedding planners help build a system simple to grasp . you organize budget and start the best prices at the minimum costs. Talk honestly about budgets. Talk about what you'll be able to produce after which you can talk to oldsters and other generous loved-ones who wish to contribute. Discuss who are assigned pay out for for what. More than anything else, it crucial to have your baby's safety to mind when buying nursery furniture sets. Guaranteed that the nursery furniture sets match the modern safety standards in your. A bassinet and a little one cradle are small scale cribs that are easily carried by a vehicle to any room the for her convenience. Babies also love the rocking and swaying features within the bassinets and baby cradles. These mini cribs are really simple to clean, fix and bring around with you, even if outdoor drives. That is the reason why most parents opt to obtain mini cribs. It would seem that Weiner has the formula for scoring an effective dramatic series down apply. First, grab a big pot throughout the kitchen. Throw into it a flawed yet compelling human . Mix in pressure, conflict, temptation, and also [http://s.zuly.de/6u4 multisitters mumbai] the world around them. Give a boil, then simmer. If your furry friend is still a puppy, you must buy pet beds that are waterproof. Luxury waterproof pet beds are lined with luxurious water-resistant fabrics and trimmed with fur. In which means you don't want to worry if your puppy wets the dog bed. Also, waterproof beds are less prone to damages unlike those will be not waterproof, which means more value to the. As I lugged the extremely heavy bed to the parking lot, I severely considered how We spent the best part of my infancy rollicking around on the cold, hard floor maybe the warm embrace of my mother. And I proved well since of course. But whats up! Just because However get appreciate the luxury of a new baby bedding is no reason for me to deprive my child! My baby will sleep in his/her own baby crib bedding.