Difference between revisions of "Successor set"

Revision as of 05:01, 26 January 2008

A set $S\subset \mathbb{R}$ is called a Successor Set iff

(i) $1\in S$

(ii) $\forall n\in S$ ; $n+1\in S$

The set of natural numbers $\mathbb{N}$ is the smallest Succesor Set because for any successor set $S$ , $\mathbb{N} \subset S$

Note that $\mathbb{N}=\{1,2,3\ldots\}$ is not the only successor set. For example, the set $S=\{1,\sqrt{2},2,1+\sqrt{2},\ldots\}$ is also a successor set.

@@ Line 5: / Line 5: @@
 (ii)<math>\forall n\in S</math>; <math>n+1\in S</math>
-Note that the set of [[Natural number|natural numbers]] <math>\mathbb{N}=\{1,2,3\ldots\}</math>is not the only successor set.
+The set of [[Natural number|natural numbers]] <math>\mathbb{N}</math> is the '''smallest''' Succesor Set because 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.
-For example, the set <math>S=\{1,\sqrt{2},2,1+\sqrt{2},\ldots\}</math> is also a successor set.
-The set <math>\mathbb{N}</math> is called the '''Smallest Succesor Set''' because for any <math>\mathbb{F}</math> that is a succesor set <math>\mathbb{N} \subset \mathbb{F}</math>

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Difference between revisions of "Successor set"

Revision as of 05:01, 26 January 2008