# Interactive Real Analysis

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## 2.3. The Principle of Induction

### Examples 2.3.2(b):

Which of the following sets are well-ordered ?
1. The number systems N, Z, Q, or R ?
2. The set of all rational numbers in [0, 1] ?
3. The set of positive rational numbers whose denominator equals 3 ?
The natural numbers N are well-ordered:
A subset of natural numbers may not have a largest element, but it must have a smallest element.

The integers Z are not well-ordered:
While many subsets of Z has a smallest element, the set Z itself does not have a smallest element.

The rationals Q are not well-ordered:
The set Q itself does not have a smallest element.

The real numbers R are not well-ordered:
R itself does not have a smallest element.

The set of all rational numbers in [0, 1] is not well-ordered:
While the set itself does have a smallest element (namely 0), the subset of all rational numbers in (0, 1) does not have a smallest element.

The set of all positive rational numbers whose denominator equals 3 is well-ordered:
This set is actually the same as the set of natural numbers, because we could simply re-label a natural number n to look like the symbol n / 3. Then both sets are the same, and hence this set is well-ordered.
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