Difference between revisions of "Successor set"
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| − | A set <math>S\subset \mathbb{R}</math> is called a ''' | + | A [[set]] <math>S\subset \mathbb{R}</math> is called a '''successor set''' [[iff]] |
| − | (i)<math>1\in S</math> | + | :(i) <math>1\in S</math> |
| − | (ii)<math>\forall n\in S</math>; <math>n+1\in S</math> | + | :(ii) <math>\forall n\in S</math>; <math>n+1\in S</math> |
| − | The set of [[ | + | The set of [[natural number]]s <math>\mathbb{N}</math> is the ''smallest'' successor set, as for any successor set <math>S</math>, <math>\mathbb{N} \subset S</math>. |
| − | Note that <math>\mathbb{N}=\{1,2,3\ldots\}</math>is not the only successor set. For example, the set <math>S=\{1,\sqrt{2},2,1+\sqrt{2},\ldots\}</math> is also a successor set. | + | Note that <math>\mathbb{N}=\{1,2,3\ldots\}</math> is not the only successor set. For example, the set <math>S=\{1,\sqrt{2},2,1+\sqrt{2},\ldots\}</math> is also a successor set. |
{{stub}} | {{stub}} | ||
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| + | [[Category:Set theory]] | ||
Latest revision as of 10:36, 26 January 2008
A set 
is called a successor set iff
- (i)
- (ii)
; 
; 
The set of natural numbers 
is the smallest successor set, as for any successor set 
, 
.
Note that 
is not the only successor set. For example, the set 
is also a successor set.
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