Download Math 320 Course Notes Chapter 7

Math 320
Course Notes
Magnhild Lien
Chapter 7: Countable and
Uncountable Sets
Hilbert’s Motel: Imagine a motel with in…nitely many rooms numbered
1; 2; 3; 4 ; n; : One evening an "in…nite" bus full with passengers drives up
to the motel asking for rooms. The seats in the bus are numbered 1; 2; 3; 4
; n;
and there is one passenger in each seat. In order to accommodate every
passenger, the motel manager tells each passenger to look at the number of his
or her seat and go to the motel room with the same number. Symbolically
he is saying: n ! n: Shortly after everyone is settled in, a mini-cooper
arrives at the motel, whose driver asks for a room for the night. The motel
manager’s …rst inclination is to say that the motel is full, but at this time, an
assistant at the motel who is also a Math 320 student at CSUN, says there is no
problem in accommodating the new arrival. Just tell the mini-cooper driver to
go to room 1 and tell all the previous hotel guests to move to the room whose
number is one greater than the number on his or her room; or symbolically
n ! n + 1: The next day after all the quests have left and all the rooms have
been cleaned, two "in…nite busses" pull into the motel parking lot. The seats on
each bus is numbered 1; 2; 3; 4 ; n;
and each passenger is requesting a room.
The manager immediately turns to the Math 320 student for help solving this
problem. What should the assistant do?
1. Bus 1: If your seat number is n you should go to room number
n
!
2. Bus 2: If your seat number is n you should go to room number
n
!
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One-to-one Correspondence
De…nition 1.1 Two sets A and B are said to have the same cardinality if
there is a bijection (one-to-one correspondence) f : A ! B: In this case we
write A B:
Example 1.2 Let’s re-examine the room assignments for the guests at
Hilbert’s Motel. For Bus 1, we seem to get a one-to-one correspondence between
the seat numbers on the bus (i.e. N) and a subset of the rooms in the Motel.
Similarly for Bus 2.
Theorem 1.3 Let A,B and C be sets, then
1. A has the same cardinality as A; that is A
A:
2. If A has the same cardinality as B; then B has the same cardinality as A;
that is A B =) B A:
3. If A has the same cardinality as B; and B has the same cardinality as
C; then A has the same cardinality as C; that is A
B and B
C
=) A C:
Proof. (part 2): If A
B, then there exists a bijection f : A ! B: Since
f is a bijection , f 1 : B ! A exists. Furthermore, if f 1 (b) = f 1 (c); then
there is an a 2 A; such that f 1 (b) = f 1 (c) = a, which implies that f (a) = b
and f (a) = c: Since f is a function this means b = c: Hence f 1 is one-to-one.
Let a 2 A; then f (a) = b for some b 2 B: If f (a) = b; then f 1 (b) = a: Hence
f 1 is onto.
How would you prove part 3?
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Remark 1.4 If S is any collection of sets, then one-to-one correspondence
is an equivalence relation on S:
Example 1.5 Any interval (a; b) has the same cardinality as R:
1. First show for any real numbers a and b; a < b, the interval (a; b) (0; 1):
De…ne f : (a; b) ! (0; 1) by f (x) = xb aa , then f is one-to-one and onto.
2. We know from trigonometry that if f (x) = tan x, then f :
is a bijection.
2; 2
!R
3. If (a; b) is an arbitrary interval, then (a; b) (0; 1) and (0; 1)
2; 2
and
R: Thus by repeated use of the transitive property, we
2; 2
conclude that (a; b) R:
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Finite and In…nite Sets
De…nition 2.1 A set A is …nite if it is the empty set or A has the same
cardinality as Sn = f1; 2; 3;
ng for some n 2 N: In this case we say the
cardinality of A is n and we write jAj = n: If A is not …nite, then we say A is
in…nite. A set A is countably in…nite if it has the same cardinality as N:
Remark 2.2 If A is countably in…nite, then there exists a bijection
f :A!N
:
Example 2.3
1. Let A = fx j x is a student in Math 320, spring 07}, then A is …nite.
De…ne a bijection f : f1; 2; 3; ; 34g ! A by
3
2. Let A = fx 2 R j x2
2 = 0g; then A is …nite and jAj =
3. Let P (x) be an n’th degree polynomial and let A = fx 2 R j P (x) = 0g.
Is A …nite? If yes, what is jAj?
Example 2.4
1. Let A = fa = 2n j n 2 Ng, then A is countably in…nite. De…ne f : A ! N
by f (a) = a2 :
2. The set of integers, Z; is countably in…nite. De…ne f : Z ! N by
f (x) =
3
2z
if
2z 1 if
z 0
z<0
Countable and Uncountable Sets
De…nition 3.1 A set is countable if it is either …nite or countably in…nite. A
set that is not countable is said to be uncountable.
Example 3.2 From the examples above, give examples of countable sets.
1.
2.
3.
4.
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Theorem 3.3 Every subset of N is countable.
See proof on page 221 in the text book.
If we construct new sets from old ones by using unions and Cartesian products, will the new sets be countable if the original sets were? YES
Theorem 3.4
1. N
N is countable.
2. If A and B are countable sets, then A
B is countable.
Proof of 1.
De…ne f : N N ! N by f (m; n) = 2m 3n : By the
Fundamental Theorem of Arithmetic, it follows that f is injective. Why?
Furthermore, since f (N N) is an in…nite subset of N; it is countably
in…nite. Hence there is a bijection g : f (N N) ! N and the composition
g f is a bijection from N N to N:
Proof of 2. If A and B are both …nite, then A B is …nite. If A and
B are both countably in…nite, then there exist bijections f : A ! N and
g : B ! N: De…ne h : A B ! N N by h(a; b) = (f (a); g(b)) then
h is a bijection. Hence A B
N N. But N N is countable, and
therefore A B is countable. Complete the proof in the case of one set
is …nite and the other is countably in…nite.
Theorem 3.5
1. If A and B are countable sets, then A [ B is countable.
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2. Let fAn gn=1 be a collection of countable sets, then B = [1
n=1 An is countable.
The proof uses many of the topics we have covered in this course and it can
be found on page 223 in the text.
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Theorem 3.6 The set Q of rational numbers is countable.
Proof: First write Q = Q+ [ f0g [ Q ; where Q+ and Q denote the sets of
positive and negative rational numbers respectively. By mapping q to q, we
see that Q+ Q : Hence if we show that Q+ is countable, and apply Theorem
3.5.1 we get that Q is countable. Let q 2 Q+ then q = ab where a and b are
positive integers with no common prime factors. De…ne f : Q+ ! N N by
f (q) = f ab = (a; b): This function is injective and hence Q+ f (Q+ ): This
completes the proof. Why?
Theorem 3.7 The set R of real numbers is uncountable.
Proof.
Since R has the same cardinality as the interval (0; 1) (Example
1.5), it su¢ ces to show that (0; 1) is uncountable. We will do this by contradiction. Suppose there is a surjection f : N ! (0; 1) ; then we can list all the
nonterminating decimals in (0; 1) in the following way
f (1) = 0:a11 a12 a13 a14
f (2) = 0:a21 a22 a23 a24
f (3) = 0:a31 a32 a33 a34
Now de…ne a number m = 0:m1 m2 m3 m4 as follows mi =
then m 2 (0; 1) and m 6= f (i) for all i 2 N:
6
2 if
1 if
aii = 1
aii 6= 1
Group Work
1. If A and B are both countably in…nite sets, then there exist bijections
f : A ! N and g : B ! N: De…ne h : A B ! N N by h(a; b) =
(f (a); g(b)) : Prove that h is a bijection.
2. Prove that the set of irrational numbers is an uncountable set.
3. Number 33 on page 224 in the text.
4. If A; B and C are sets, prove: If A has the same cardinality as B; and B
has the same cardinality as C; then A has the same cardinality as C; that
is A B and B C =) A C:
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