Page 336 - HOW TO PROVE IT: A Structured Approach, Second Edition

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322 Infinite Sets
of English words. First show that the set of all grammatical sentences is
countable, and then show that it is infinite.)
11. Some real numbers can be defined by a phrase in the English language.
For example, the phrase “the ratio of the circumference of a circle to its
diameter” defines the number π.
(a) Prove that the set of numbers that can be defined by an English phrase
is denumerable. (Hint: See exercise 10.)
(b) Prove that there are real numbers that cannot be defined by English
phrases.

7.3. The Cantor–Schr¨oder–Bernstein Theorem

Suppose A and B are sets and f is a one-to-one function from A to B. Then f
shows that A ∼ Ran( f ) ⊆ B, so it is natural to think of B as being at least as
large as A. This suggests the following notation:

Definition 7.3.1. If A and B are sets, then we will say that B dominates A, and
write A B, if there is a function f : A → B that is one-to-one. If A B
and A ∼ B, then we say that B strictly dominates A, and write A ≺ B.

For example, in the proof of Theorem 7.2.6 we gave a one-to-one function
f : P (Z ) → R,so P (Z ) R. Of course, for any sets A and B,if A ∼ B
+
+
then also A B. It should also be clear that if A ⊆ B then A B. For example,
Z R. In fact, by Theorem 7.2.6 we also know that Z + ∼ R, so we can say
+
that Z + ≺ R.
You might think that would be a partial order, but it turns out that it isn’t.
You’re asked in exercise 1 to check that is reflexive and transitive, but it
is not antisymmetric. (In the terminology of exercise 24 of Section 4.6, is
a preorder.) For example, Z ∼ Q,so Z Q and Q Z , but of course
+
+
+
+
Z = Q. But this suggests an interesting question: If A B and B A, then
A and B might not be equal, but must they be equinumerous?
The answer, it turns out, is yes, as we’ll prove in our next theorem. Several
mathematicians’ names are usually associated with this theorem. Cantor proved
a limited version of the theorem, and later Ernst Schr¨oder (1841–1902) and
Felix Bernstein (1878–1956) discovered proofs independently.

Theorem 7.3.2. (Cantor–Schr¨oder–Bernstein theorem) Suppose A and B are
sets. If A B and B A, then A ∼ B.