Numbers - Revision history

Weihe: /* Natural numbers */

2015-06-09T18:09:18Z

Natural numbers

← Older revision		Revision as of 18:09, 9 June 2015
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	'''Induction:'''		'''Induction:'''
	[http://en.wikipedia.org/wiki/Mathematical_induction Mathematical induction] is a well-known method to prove a statement <math>S(n)</math> for all natural numbers <math>n\geq n_S</math>, where <math>n_S</math> is some integral number specific to ´<math>S</math>. Two types of induction are often distinguished:		[http://en.wikipedia.org/wiki/Mathematical_induction Mathematical induction] is a well-known method to prove a statement <math>S(n)</math> for all natural numbers <math>n\geq n_S</math>, where <math>n_S</math> is some integral number specific to <math>S</math>. Two types of induction are often distinguished:
	# Ordinary induction: for <math>n>n_S</math>, <math>S(n-1)</math> is the induction hypothesis for the proof of <math>S(n)</math>.		# Ordinary induction: for <math>n>n_S</math>, <math>S(n-1)</math> is the induction hypothesis for the proof of <math>S(n)</math>.
	# Strong induction: for <math>n>n_S</math>, <math>S(n_S)\wedge S(n_S+1)\wedge\ldots\wedge S(n-1)</math> is the induction hypothesis for the proof of <math>S(n)</math>.		# Strong induction: for <math>n>n_S</math>, <math>S(n_S)\wedge S(n_S+1)\wedge\ldots\wedge S(n-1)</math> is the induction hypothesis for the proof of <math>S(n)</math>.

Weihe: /* Natural numbers */

2015-06-09T18:08:55Z

Natural numbers

← Older revision		Revision as of 18:08, 9 June 2015
Line 11:		Line 11:

	'''Induction:'''		'''Induction:'''
	[http://en.wikipedia.org/wiki/Mathematical_induction Mathematical induction]] is a well-known method to prove a statement <math>S(n)</math> for all natural numbers <math>n\geq n_S</math>, where <math>n_S</math> is some integral number specific to ´<math>S</math>. Two types of induction are often distinguished:		[http://en.wikipedia.org/wiki/Mathematical_induction Mathematical induction] is a well-known method to prove a statement <math>S(n)</math> for all natural numbers <math>n\geq n_S</math>, where <math>n_S</math> is some integral number specific to ´<math>S</math>. Two types of induction are often distinguished:
	# Ordinary induction: for <math>n>n_S</math>, <math>S(n-1)</math> is the induction hypothesis for the proof of <math>S(n)</math>.		# Ordinary induction: for <math>n>n_S</math>, <math>S(n-1)</math> is the induction hypothesis for the proof of <math>S(n)</math>.
	# Strong induction: for <math>n>n_S</math>, <math>S(n_S)\wedge S(n_S+1)\wedge\ldots\wedge S(n-1)</math> is the induction hypothesis for the proof of <math>S(n)</math>.		# Strong induction: for <math>n>n_S</math>, <math>S(n_S)\wedge S(n_S+1)\wedge\ldots\wedge S(n-1)</math> is the induction hypothesis for the proof of <math>S(n)</math>.

Weihe: /* Natural numbers */

2015-06-09T18:08:44Z

Natural numbers

← Older revision		Revision as of 18:08, 9 June 2015
Line 11:		Line 11:

	'''Induction:'''		'''Induction:'''
	[http://en.wikipedia.org/wiki/Mathematical_induction ~~Matthematical~~ induction]] is a well-known method to prove a statement <math>S(n)</math> for all natural numbers <math>n\geq n_S</math>, where <math>n_S</math> is some integral number specific to ´<math>S</math>. Two types of induction are often distinguished:		[http://en.wikipedia.org/wiki/Mathematical_induction Mathematical induction]] is a well-known method to prove a statement <math>S(n)</math> for all natural numbers <math>n\geq n_S</math>, where <math>n_S</math> is some integral number specific to ´<math>S</math>. Two types of induction are often distinguished:
	# Ordinary induction: for <math>n>n_S</math>, <math>S(n-1)</math> is the induction hypothesis for the proof of <math>S(n)</math>.		# Ordinary induction: for <math>n>n_S</math>, <math>S(n-1)</math> is the induction hypothesis for the proof of <math>S(n)</math>.
	# Strong induction: for <math>n>n_S</math>, <math>S(n_S)\wedge S(n_S+1)\wedge\ldots\wedge S(n-1)</math> is the induction hypothesis for the proof of <math>S(n)</math>.		# Strong induction: for <math>n>n_S</math>, <math>S(n_S)\wedge S(n_S+1)\wedge\ldots\wedge S(n-1)</math> is the induction hypothesis for the proof of <math>S(n)</math>.

Weihe: /* Natural numbers */

2015-06-09T18:08:30Z

Natural numbers

← Older revision		Revision as of 18:08, 9 June 2015
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	We say that <math>\mathbb{N}_{0}</math> is the set of all '''natural numbers'''.		We say that <math>\mathbb{N}_{0}</math> is the set of all '''natural numbers'''.

			'''Induction:'''
			[http://en.wikipedia.org/wiki/Mathematical_induction Matthematical induction]] is a well-known method to prove a statement <math>S(n)</math> for all natural numbers <math>n\geq n_S</math>, where <math>n_S</math> is some integral number specific to ´<math>S</math>. Two types of induction are often distinguished:
			# Ordinary induction: for <math>n>n_S</math>, <math>S(n-1)</math> is the induction hypothesis for the proof of <math>S(n)</math>.
			# Strong induction: for <math>n>n_S</math>, <math>S(n_S)\wedge S(n_S+1)\wedge\ldots\wedge S(n-1)</math> is the induction hypothesis for the proof of <math>S(n)</math>.
			Both types of induction are logically equivalent. Therefore, in this wiki, they are not distinguished from each other: induction always means strong induction.

	==Real numbers==		==Real numbers==

Cuozzo at 09:41, 1 October 2014

2014-10-01T09:41:50Z

@@ Line 1: / Line 1: @@
 [[Category:Background]]
 ==Natural numbers==
@@ Line 8: / Line 9: @@
 We say that <math>\mathbb{N}_{0}</math> is the set of all '''natural numbers'''.

Cuozzo at 17:45, 30 September 2014

2014-09-30T17:45:49Z

← Older revision		Revision as of 17:45, 30 September 2014
Line 8:		Line 8:

	We say that <math>\mathbb{N}_{0}</math> is the set of all '''natural numbers'''.		We say that <math>\mathbb{N}_{0}</math> is the set of all '''natural numbers'''.

	~~===Known Related Topics===~~
	~~===Remark===~~
	~~===Reference===~~

Cuozzo: Created page with "Category:Background ==Natural numbers== $\mathbb{N}$ denotes the set of positive integral numbers: : $\mathbb{N} := \{1,2,3,...\}$ : $\mathbb{..."$

2014-09-30T17:39:21Z

Created page with "Category:Background ==Natural numbers== <math>\mathbb{N}</math> denotes the set of positive integral numbers: : <math>\mathbb{N} := \{1,2,3,...\}</math> : <math>\mathbb{..."

New page

[[Category:Background]]
==Natural numbers==
<math>\mathbb{N}</math> denotes the set of positive integral numbers:

: <math>\mathbb{N} := \{1,2,3,...\}</math>

: <math>\mathbb{N}_{0} := \mathbb{N} \cup \{0\}</math>

We say that <math>\mathbb{N}_{0}</math> is the set of all '''natural numbers'''.

===Known Related Topics===
===Remark===
===Reference===

Numbers - Revision history

Weihe: /* Natural numbers */

Weihe: /* Natural numbers */

Weihe: /* Natural numbers */

Weihe: /* Natural numbers */

Cuozzo at 09:41, 1 October 2014

Cuozzo at 17:45, 30 September 2014

Cuozzo: Created page with "Category:Background ==Natural numbers== \mathbb{N} denotes the set of positive integral numbers: : \mathbb{N} := \{1,2,3,...\} : \mathbb{..."

Cuozzo: Created page with "Category:Background ==Natural numbers== $\mathbb{N}$ denotes the set of positive integral numbers: : $\mathbb{N} := \{1,2,3,...\}$ : $\mathbb{..."$