## 2.3. The Principle of Induction

### Examples 2.3.2(b):

Which of the following sets are well-ordered ?

- The number systems
**N**,**Z**,**Q**, or**R**? - The set of all rational numbers in [0, 1] ?
- 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.