Page 329 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Countable and Uncountable Sets 315
23. Suppose A is a finite set and R is an equivalence relation on A.
Suppose also that there is some positive integer n such that ∀x ∈
A(|[x] R |= n). Prove that A/R is finite and |A/R|=|A|/n. (Hint: Use
exercise 19.)
24. (a) Suppose that A and B are finite sets. Prove that A ∪ B is finite, and
|A ∪ B|=|A|+|B|−|A ∩ B|.
(b) Suppose that A, B, and C are finite sets. Prove that A ∪ B ∪ C is
finite, and |A ∪ B ∪ C|=|A|+|B|+|C|−|A ∩ B|−|A ∩ C|−
|B ∩ C|+|A ∩ B ∩ C|.
∗ 25. In this problem you will prove the Inclusion–Exclusion Principle, which
generalizes the formulas in exercise 24. Suppose A 1 , A 2 , ..., A n are finite
sets. Let P = P (I n )\ {∅}, and for each S ∈ P let A S =∩ i∈S A i . Prove

|S|+1

A i = (−1) |A S |. (The notation
S∈P
that ∪ i∈I n A i is finite and ∪ i∈I n
on the right side of this equation denotes the result of running through
all sets S ∈ P, computing the number (−1) |S|+1 |A S | for each S, and then
adding these numbers. Hint: Use induction on n.)
26. Prove that if A and B are finite sets and |A|=|B|, then |A B| is even.
27. Each customer in a certain bank has a PIN number, which is a sequence
of four digits. Show that if the bank has more than 10,000 customers,
then some two customers must have the same PIN number. (Hint: See
exercise 11.)
28. Alice opened her grade report and exclaimed, “I can’t believe Professor
Jones flunked me in Probability.” “You were in that course?” said Bob.
“That’s funny, I was in it too, and I don’t remember ever seeing you there.”
“Well,” admitted Alice sheepishly, “I guess I did skip class a lot.” “Yeah,
me too” said Bob. Prove that either Alice or Bob missed at least half of
the classes.

7.2. Countable and Uncountable Sets

Often when we perform some set-theoretic operation with countable sets, the
result is again a countable set.

Theorem 7.2.1. Suppose A and B are countable sets. Then:

1. A × B is countable.
2. A ∪ B is countable.

Proof. Since A and B are countable, by Theorem 7.1.5 we can choose one-to-
+
one functions f : A → Z and g : B → Z .
+