Math 220 Workshop 6: Cardinality II, and review
1. Recall the notation: P(A) is the power set of A. Recall the theorem we started proving in the last
class: |P(A)| > |A|. The goal of this problem is to finish the proof we started working on: we need
to prove that there is no bijective function from A to P(A). We prove it by contradiction. Suppose
such a bijective function f existed. That is, suppose we had a bijective function f : A → P(A). Then
consider the set S = {x ∈ A : x ∈
/ f (X)}. Then S ⊂ A, so S ∈ P(A). Prove that f −1 (S) is empty.
(Hint: use contradiction again. Suppose there was some element a ∈ A such that f (a) = S. Then
ponder the question: does a belong to S?) Complete the proof of the theorem that |P(A)| > |A|.
2. Definition. Let A be an arbitrary set and let B be a subset of A. Define the function χB : A → {0, 1}
by the formula
(
0 if x ∈
/ B,
χB (x) :=
1 if x ∈ B.
This function is called the characteristic function of B.
(a) Draw the graph of the characteristic function of (0, 1) as a subset of R.
(b) Does there exist a non-empty subset of R such that its characteristic function is continuous
everywhere?
3. Let P be the set of all possible sequences of 0’s and 1’s. Find a very natural bijective function between
P and P(N).
(Hint: construct a function from P(N) to P , whose definition uses the notion of the characteristic
function).
4. Let P be the set of all possible sequences of 0’s and 1’s, as above. Prove that P is uncountable. Try
to see the similarity between the proof of this statement and the proof discussed in Problem 1.
5. Prove that the cardinality of the set P from the previous problem is, in fact, continuum, that is, there
is a bijective function from P to R (this fact is used without proof in Problem 9 of the last Homework).
6. What is wrong with the following sketch of a proof that |R × R| = |R|:
let R = {a1 , . . . , an , . . . }. Then R × R is the set of pairs (ai , bj ) where i, j ∈ N, and now we can
arrange these pairs in a table, and use the algorithm for traversing the table to put them into one long
list {c1 , . . . cn , . . . }.
(Note that this proof works fine for N × N and Q × Q).
Review Problems
7. Prove that for any n,
n
X
k=0
8. Let A be the subset of N × N defined by
qi =
q n−1 − 1
.
q−1
A = {(a, b) : 3|a and 5|b}.
Let f : A → N be the function defined by f ((a, b)) = a + b. Find the range of f .
9. (a) Show that every simple graph has two vertices of the same degree.
(b) Show that if n people attend a party and some shake hands with others (but not with themselves), then at the end, there are at least two people who have shaken hands with the same
number of people.
10. A graph with no cycles is called a forest. Show that if a forest has v vertices, e edges, and c connected
components, then v = e + c.