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Chapter 2
Section 2.5
Infinite Sets
Sets and One-to-One Correspondences
An important tool the mathematicians use to compare the size of sets is called a
one-to-one correspondence. This concept is a way of saying two sets are the same
size without counting the numbers in them. We call two sets equivalent if they have
the same number of elements. Equivalent sets can be put into one-to-one
correspondence with each other by showing how all the elements of one set exactly
match with all the elements of another set. You can represent different one-to-one
correspondences by drawing arrows between the sets.
January
Larry
January
Larry
June
Curly
June
Curly
July
Moe
July
Moe
Each of the illustrations above shows a one-to-one correspondence between the
sets {January, June , July} and {Larry, Curly, Moe}. These two sets would be
considered equivalent but not equal.
Can the set {January, June, July} be put into one-to-one correspondence with the
set {Red, Green, Blue, Orange}? NO !
Sets that are equal have exactly the same elements in them. Sets that are
equivalent need only have the same number of elements in them.
The sets {January, June, July} and {Red, Green, Blue} are equivalent but not
equal. The sets {January, June, July} and {July, June, January} are both equal and
equivalent.
Often times it is useful to draw or picture the one-to-one correspondence in row
format. For example a one-to-one correspondence between the sets {January,
June, July} and {Red, Green, Blue} can be illustrated as in the figure below:
{January,
June,
July}
↕
↕
↕
{Red,
Green,
Blue}
Finite vs. Infinite Sets
A set is finite if it can be put in one-to-one correspondence with a set of the form
{1,2,3,4,…,n} for some number n.
Example:
The set {,,,} is finite because it can be
put into one-to-one correspondence with the
set {1,2,3,4} as shown to the right. (i.e. n=4)
{,
,
,
}
↕
↕
↕
↕
{1,
2,
3,
4}
A set is called infinite if it can be put into one-to-one correspondence with a
proper subset of itself. We use this idea rather than "going on forever" because it
relates back to sets. This gives us a tool to work with infinite sets.
Example:
the set of natural numbers is infinite. We are going to show a one-to-one
correspondence between N={1,2,3,…} and {2,3,4,…}. Notice that {2,3,4}N (i.e.
{2,3,4} is a proper subset of the set N)
Find x, y and z
N = {1,
x,
…, 287, …,
z,
↕
↕
↕
↕
4,
…, 59, …,
y,
2,
3,
↕
↕
{2,
3,
…,
…, 311,
…}
…}
x = 58
y = 288
z = 310
N = {1,
↕
D = {1,
…,
…,
2,
3,
↕
↕
↕
↕
3,
5,
…, 21, …,
y,
x,
58,
…,
z,
…} Find x, y and z
↕
…, 133,
x = (21+1)2 = 11
…}
y = 58·2 -1= 115
z = (133+1)2 = 67
n
To get from a number in N to a

number in D we double the
number then minus 1.
2n  1
N = {1,
↕
2,
3,
↕
↕
…,
To get from a number in D to
a number in N we add 1 then
divide by 2.
z,
↕
↕
B = {11, 12, 13, …, 25, …,
y,

n
Find x, y and z
x, …, 96, …,
↕
n 1
2
…, 211,
...}
…}
x = 25-10 =15
y = 96 +10 = 106
z = 211 -10 = 201
To get from a number in
N to a number in B we
add 10.
n

n  10
To get from a number in
B to a number in N we
subtract 10.
n  10

n
Sets in One-to-One Correspondence with N
We will see there are many different sets that are in one-to-one correspondnece
with the set of natural numbers N. One of the most famous examples is the set of
integers I. This seems very counter intuitive since there seems like there should
be "twice" as many integers as natural numbers, but this is not the case.
There are just as many integers as natural numbers!
We show the one-to-one correspondence below by "interweaving" the positive and
negative numbers.
34 ,35
N=
I=
…,
46
…,
{1,
2,
3,
4,
5,
6,
7,
↕
↕
↕
↕
↕
↕
↕
↕
↕
{0
1,
-1,
2,
-2,
3,
-3,
…, -17, …,
23,
x,
17
y,
86
…,
87,
…}
↕
…,
z,
…}
43,-43
For a negative number in I look just ahead of it, find the positive and multiply by 2.
For a positive number in I multiply by 2.
For an even number in N divide by 2.
For an odd number in N divide look at the even ahead divide by 2 and the next
one will be negative.
1
1
1
2
1
3
1
4
1
5
1
6
1
7
2
1
2
2
2
3
2
4
2
5
2
6
3
1
3
2
3
3
3
4
3
5
3
6
2
7
3
7
4
1
4
2
4
3
4
4
4
5
4
6
4
7
5
1
5
2
5
3
5
4
5
5
5
6
5
7
6
1
6
2
6
3
6
4
6
5
6
6
6
7
7
1
7
2
7
3
7
4
7
5
7
6
7
7
{1,
2,
3,
4,
5,
6,
↕
↕
↕
↕
↕
↕
N=
We line up the fractions
with the same numerators
going across and the same
denominators going down.
We then serpentine back
and forth skipping over any
unreduced fraction.
…}
Q+ =
{1,
1
2
,
2,
3,
1
3
,
1
4
,
…}
N=
{1,
2,
3,
4,
5,
6,
…}
↕
↕
↕
↕
↕
↕
{0,
1,
-1,
1
2
Q=
,
1
2
,
2,
…}
A Different Infinity
The closed interval [0,1] (i.e. every number that can be written as a decimal
between 0 and 1) is called the unit interval.
The unit interval [0,1] is not equivalent to N.
In other words the infinity represented by the natural numbers is a different type of
infinity that is represented by the unit interval [0,1]. The reasoning for this is very
ingenious.
Suppose the unit interval [0,1] has a one-to-one correspondence with N. We don't
know what numbers in [0,1] correspond to N so we call them x1, x2, x3,….
N=
{1,
2,
3,
4,
5,
↕
↕
↕
↕
↕
x2,
x3,
x4,
x5,
[0,1] = {x1,
…}
…}
If we knew
the numbers
arrange them
this way.
We can always create a number that is not in this list
by changing the digit in red to a 4 if it is not a 4 and to
a 5 if it is a 4. In this case the new number would be:
0.4544…
N
[0,1]
1
↔ 0.132786…
2
↔ 0.345802…
3
↔ 0.035211…
4
↔ 0.250000…


Closed Intervals
If we start with any closed interval [a,b] we can show it has just as many points
(i.e. can be put into one-to-one correspondence) with the unit interval [0,1]. This
can be visualized as making the endpoints match up from a common point. For
example if we want to show the closed interval [3,7] is equivalent to the closed unit
interval [0,1] we show how the points correspond.
To locate the points that correspond to x and
y on the other interval we first locate point P
P
0
3
y 1
7
x
P
0
3
We then draw a straight line connecting P
and x or y. Where that line hits the other
segment is the corresponding point.
This could also be calculated directly if you
knew one of the numbers by using a
proportion.
y 1
6
3
4
7
y 3

1 4
3
y
4
Equivalent Shapes
Just like two segments of different size represent the same infinity so can different
shapes. For example the are just as many points on the small circle below as
there are on the large rectangle.
To find the points that correspond to the
orange, green and blue points draw a line
from the black point. Where it hits the other
shape is the corresponding point.
2
It is also a well know fact that the unit
segment [0,1] is equivalent to the unit
square [0,1]  [0,1]. This is an amazing
fact because these two shapes are of
different dimension. A line segment is 1
dimensional where the unit square is two
dimensional.
1.5
1
0.5
-0.5
0
0.5
1
1.5
2
-0.5
0.5
1
-0.5
These two sets are equivalent!
1.5
2
Reference Sets
A reference set for a number is any set that has that number of elements in it. For
example all of the sets listed below are reference sets for the number 4. Putting a
set inside | | refers to its cardinal number (i.e. |{a, b, c, d}| = 4).
{red, green, blue, orange}
{,,,}
Cardinal Numbers
A cardinal number is the collection of all sets
that are equivalent to a particular reference set.
Below is a table of cardinal numbers and a
reference set for each one.
{spring, summer, fall, winter}
Cardinal
Number
Reference Set
0

1
{a}
2
{one, won}
3
{,,}
The symbol c is used to represent the number of
elements in the set [0,1] the unit interval. This is
called the cardinality of the continuum.
4
{,,,}
The cardinal number that a set belongs to is
called the cardinality of the set. For example the
cardinality of {,,} is 3.
0
The symbol 0 (pronounced aleph null) is used
to represent the number of elements in a set
equivalent to N the natural numbers.

c

N = {1,2,3,…}
[0,1]