Difference between revisions of "Natural number"

Revision as of 02:55, 26 January 2008

The set of natural numbers, denoted $\mathbb{N}$ , is the set most conveniently associated with the notion of "counting".

Definition

Let $\mathcal{F}$ be the set of all successor sets $S$ .

The set of Natural Numbers $\mathbb{N}$ is defined as $\mathbb{N}=\bigcap_{S\in\mathcal{F}} S$

Note that as $1\in S$ $\forall S\in\mathcal{F}$ , $\mathbb{N}$ is non-empty.

Common Usage

According to this definition, $\mathbb{N}$ is the set $\{1,2,3,\ldots\}$ (Which is also called the set of counting numbers or positive integer)s. Unfortunately, in some texts, $\mathbb{N}$ is taken to be the set of whole numbers or nonnegative integers. Because of this ambiguity, one should always be careful to define one's notation clearly. Possible alternatives include $\mathbb{Z}_{\geq0}$ for the non-negative integers and $\mathbb{Z}_{>0}$ or $\mathbb{P}$ for the positive integers (although $\mathbb{P}$ is also sometimes used for the prime numbers). Natural numbers are important in the link between the well-ordering principle and the principle of mathematical induction.

Mathematical Induction

Mathematical Induction is an extremely useful tool for problems regarding Natural Numbers.

Statement:

Let $S\subset \mathbb{N}$

Let (i) $1\in S$

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

Then $S$ is the set of natural numbers, or $S=\mathbb{N}$

Well-Ordering Principle

The Well-Ordering Principle states that every subset of $\mathbb{N}$ has a least element.

@@ Line 1: / Line 1: @@
-The set of '''natural numbers''', denoted <math>\mathbb{N}</math>, is a subset of the [[integer]]s <math>\mathbb{Z}</math>.  Unfortunately, exactly which subset is not entirely clear: in some texts, <math>\mathbb{N}</math> is taken to be the set of [[counting number]]s ([[positive integer]]s), while in others it is taken to be the set of [[whole number]]s ([[nonnegative]] integers).  Because of this ambiguity, one should always be careful to define one's notation clearly.  Possible alternatives include<math>\mathbb{Z}_{\geq0}</math> for the non-negative integers and  <math>\mathbb{Z}_{>0}</math> or <math>\mathbb{P}</math> for the positive integers (although <math>\mathbb{P}</math> is also sometimes used for the [[prime number]]s).
+The set of '''natural numbers''', denoted <math>\mathbb{N}</math>, is the set most conveniently associated with the notion of "counting".
+==Definition==
+Let <math>\mathcal{F}</math> be the set of all [[Successor set |successor sets]] <math>S</math>.
+The set of Natural Numbers <math>\mathbb{N}</math> is defined as
+<math>\mathbb{N}=\bigcap_{S\in\mathcal{F}} S</math>
+Note that as <math>1\in S</math> <math>\forall S\in\mathcal{F}</math>, <math>\mathbb{N}</math> is non-empty.
+==Common Usage==
+According to this definition, <math>\mathbb{N}</math> is the set <math>\{1,2,3,\ldots\}</math> (Which is also called the set of [[counting number]]s or [[positive integer]])s. Unfortunately, in some texts, <math>\mathbb{N}</math> is taken to be the set of [[whole number]]s or [[nonnegative]] integers.  Because of this ambiguity, one should always be careful to define one's notation clearly.  Possible alternatives include<math>\mathbb{Z}_{\geq0}</math> for the non-negative integers and  <math>\mathbb{Z}_{>0}</math> or <math>\mathbb{P}</math> for the positive integers (although <math>\mathbb{P}</math> is also sometimes used for the [[prime number]]s).
 Natural numbers are important in the link between the well-ordering principle and the principle of mathematical induction.
+==Mathematical Induction==
+'''Mathematical Induction''' is an extremely useful
+tool for problems regarding Natural Numbers.
+Statement:
+Let <math>S\subset \mathbb{N}</math>
+Let (i)<math>1\in S</math>
+Let (ii)<math>\forall n\in S</math>; <math>n+1\in S</math>
+Then <math>S</math> is the set of natural numbers, or <math>S=\mathbb{N}</math>
+==Well-Ordering Principle==
+The '''Well-Ordering Principle''' states that every subset of <math>\mathbb{N}</math> has a least element.
 [[Category:Definition]]
 [[Category:Number theory]]

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