Discrete Mathematics

----------------------------COMBINATORIC$

265

If

either

or

both sets A and B are finite, omit from the listing all pairs containing non-existent elements

a

orb.

Note that the ordering

of

elements in this

proof

corresponds to the ordering

in

Fig. 8.3. In fact, the way

I }

we

showed that the positive rationals are countable was a special case

of

the above proof.

8.11

THE

CONTINUUM

The real numbers, R, are sometimes referred to as the continuum.

Every real number is the least upper bound

of

the set

of

rationals less than it that is, lub : P (Q) --7 R

is surjection, and therefore R inj P (Q). But

we

have seen that N bij Q, which implies P (N) ...