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| 4) |
Completeness property |
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Geometrically, set of all points on a line represent the set of all real numbers. There are some special subsets of  which are important. These are the familiar number systems. |
| 1.1.2 |
 , the set of Natural Numbers: |
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Recall that, there exist unique elements 0, 1  such that  , for x and , for all 0  x .One can show that . The set  is the 'smallest' subset of having the property : and  , whenever n . This is also called the Principle of Mathematical Induction. One can show that such a subset of exists, and is unique. Elements of are called natural numbers. We shall use the familiar notation, = {1,2,....,}. Geometrically, we can select any arbitrary point O on the real line and associate it with 0 . Equidistant points on the right of O can be labeled as  |
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The set has the following properties, which we shall assume:
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 for all  . |
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For every  . |
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For every , there is no element m such that  . |
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Archimedean property |
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For every  , there exists such that . |
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| 1.1.3 |
Definition: |
(i)
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A subset   is said to be bounded above if there exists  such that r  for all r E. That is, all the elements of E lie to the left of s, up to s at most. |
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