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

June

July

Larry

Curly

Moe

January

June

July

Larry

Curly

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}?

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.

NO !

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)

N = {1, 2, 3, …, x, …, 287, …, z, …}

↕ ↕ ↕ ↕ ↕ ↕

{2, 3, 4, …, 59, …, y, …, 311, …}

x = 58

y = 288

z = 310

Find x, y and z

N = {1, 2, 3, …, x, …, 58, …, z, …}

↕ ↕ ↕ ↕ ↕ ↕

D = {1, 3, 5, …, 21, …, y, …, 133, …}

x = (21+1)2 = 11

y = 58·2 -1= 115

z = (133+1)2 = 67

Find x, y and z

To get from a number in N to a number in D we double the number then minus 1.

To get from a number in D to a number in N we add 1 then divide by 2.

N = {1, 2, 3, …, x, …, 96, …, z, ...}

↕ ↕ ↕ ↕ ↕ ↕

B = {11, 12, 13, …, 25, …, y, …, 211, …}

x = 25-10 =15

y = 96 +10 = 106

z = 211 -10 = 201

Find x, y and z

To get from a number in N to a number in B we add 10.

To get from a number in B to a number in N we subtract 10.

12 n

n

n

n

21

10n

n

n

n

10

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.

N = {1, 2, 3, 4, 5, 6, 7, …, x, …, y, …, 87, …}

↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕

I = {0 1, -1, 2, -2, 3, -3, …, -17, …, 23, …, z, …}

17

34

For a negative number in I look just ahead of it, find the positive and multiply by 2.

,35 46 86

For a positive number in I multiply by 2.

43,-43

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

12

1

1

2

1

3

1

4

1

5

1

6

1

7

2

2

3

1

4

1

5

16

17

1

3

2

4

25

2

6

2

7

2

2

3

2

4

3

3

4

3

5

36

37

3

3

4

4

45

4

6

4

7

4

2

5

2

6

3

5

4

5

5

56

57

5

3

6

4

65

6

6

6

7

6

2

73

7

4

75

7

6

7

7

7

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.

N = {1, 2, 3, 4, 5, 6, …}

↕ ↕ ↕ ↕ ↕ ↕

Q+= {1, , 2, 3, , , …}2

13

1

4

1

N = {1, 2, 3, 4, 5, 6, …}

↕ ↕ ↕ ↕ ↕ ↕

Q= {0, 1, -1, , , 2, …}2

12

1

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, …}

↕ ↕ ↕ ↕ ↕

[0,1] = {x1, x2, x3, x4, x5, …}

N [0,1]

1 ↔ 0.132786…

2 ↔ 0.345802…

3 ↔ 0.035211…

4 ↔ 0.250000…

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…

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.

0 1

3 7x

y

To locate the points that correspond to x and y on the other interval we first locate point P

We then draw a straight line connecting P and x or y. Where that line hits the other segment is the corresponding point.

P

0 1

3 7

y

P This could also be calculated directly if you knew one of the numbers by using a proportion.

6

4

3

4

34

3

1

y

y

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.

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.

-0.5 0.5 1 1.5 2

-0.5

0.5

1

1.5

2

-0.5 0 0.5 1 1.5 2

These two sets are equivalent!

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} {,,,} {spring, summer, fall, winter}

Cardinal

Number

Reference Set

0

1 {a}

2 {one, won}

3 {,,}

4 {,,,}

0 N = {1,2,3,…}

c [0,1]

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.

The symbol 0 (pronounced aleph null) is used to represent the number of elements in a set equivalent to N the natural numbers.

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.

The cardinal number that a set belongs to is called the cardinality of the set. For example the cardinality of {,,} is 3.