UIUC Math 347H Lecture 9: Discussion questions
Countability and other basic ideas. This section contains a lot
of very interesting theory and in particular some classic, non-trivial
theorems. I consider the entire section worthy of careful reading.
Key definitions:
• A set A is finite if it is empty or is in bijection with [n] for
some n.
• A set is infinite if it is not finite.
• An (infinite set) has cardinality “aleph null”, denoted ℵ0 if it is
in bijection with N
1. We will discuss the main definitions above.
2. We will go over the proof of Schröder-Bernstein: If A and B are
sets (possibility infinite!) and there are injections f : A → B and
g : B → A then there is a bijection between A and B.
This is a tricky proof (go look at the history of it). I will likely pause
to have you ponder a key point or two.
3. I will discuss the “sloppy” proof that Q+ is in bijection with N.
Hence the cardinality of Q+ is ℵ0 .
4. To give our second proof of the conclusion of Q3, we will go over Exn ···a2 a1
ample 1.7.14(iv): We define a map f : Q+ → N by sending ab = abm
···b2 b1
(in lowest terms, expanded in base 10) to the number an · · · a1 dbm · · · b1
in base 11 (which uses the digits 0, 1, . . . , 9, d). Is this an injection, surjection, bijection?
Now how do you use Schröder-Bernstein to reprove Q3?
5. Prove every infinite set has a countably infinite subset.
6. Are there sets “bigger” than N? Yes: we prove Theorem 1.8.27: If A
2
is any set (including the empty set) then there is no bijection between
A and Powerset(A).
(i) Why is it true when A = ∅?
(ii) This is a Russell’s paradox like tongue-twister. Suppose there is a
bijection f : A → PowersetA. Let Pa = f (a). Now let B = {a|a 6 inPa }.
Let b = f −1 (B) and hence b = Pb . Is b ∈ B? If b ∈ B then by definition
of B, b 6∈ Pb . However, B = Pb , so b 6∈ B. Thus b ∈ Pb = B, a
contradiction.
Define c to be the cardinality of Powerset(A) where A is any countably infinite set.
7. Exercise 1.8.29: Prove the definition of c is independent of A: that
is if A and B are countably infinite then there is a bijection between
Powerset(A) and Powerset(B).
8. We prove Theorem 1.8.32: the set of all real numbers between 0
and 1 is not countable. Here’s the main idea. If it were countable, line
them up:
a1 = 0.a1,1 a1,2 · · · a1,n · · ·
a2 = 0.a2,1 a2,2 · · · a2,n · · ·
···
am = 0.am,1 am,2 · · · am,n · · ·
···
Now construct b = 0.b1 b2 · · · by picking bi to differ from ai,i for each i
– a technicality: don’t pick 9 ever. As long as we do no allow repeating
9’s, every decimal expansion is unique and so b is not equal to anything
else on our list, a contradiction.