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.