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Counting Sets

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Counting Sets

- A function is called one-to-one if every element of B is mapped to at most one element of A.

b1

a1

A

B

a2

b2

a3

b3

b4

- A function is called onto if every element of B is mapped to at least one element of A.

b1

a1

A

B

a2

b2

a3

b3

a4

- If a function is both one-to-one and onto then every element of B is mapped to exactly one element of A.

a1

b1

A

B

a2

b2

a3

b3

- Two sets A, B are of the same size if they have an equal number of elements. We denote this as |A| = |B|. There is an 1-1 and onto function mapping each element of A to an element of B iff |A| = |B|.

- We say that two infinite sets A, B are of the same size if there is an one to one and onto function from A to B (or from B to A)

|A| = |B|

a1

b1

A

B

a2

b2

a3

b3

…

…

- A set is countable if you can find an one to one and onto correspondence with the natural numbers (intuitively this means that it has the same size as the natural numbers)

1

a

2

b

A

N

3

c

4

d

5

e

…

…

- There is an one to one an onto function from E (the even numbers) to N: f(n) = n/2

f

0

0

1

2

N

2

4

E

3

6

4

8

…

…

. . .

(0,0)

(0,1)

(0,2)

(0,3)

(0,4)

(1,0)

(1,1)

(1,2)

(1,3)

(1,4)

(2,0)

(2,1)

(2,2)

(2,3)

(2,4)

(3,0)

(3,1)

(3,2)

(3,3)

(3,4)

(4,0)

(4,1)

(4,2)

(4,3)

(4,4)

.

.

.

. . .

. . .

0

1

8

9

24

3

2

7

10

23

5

6

4

11

22

15

14

13

12

21

17

18

19

20

16

.

.

.

. . .

. . .

0/0

0/1

0/2

0/3

0/4

1/0

1/1

1/2

1/3

1/4

2/0

2/1

2/2

2/3

2/4

3/0

3/1

3/2

3/3

3/4

4/0

4/1

4/2

4/3

4/4

.

.

.

. . .

. . .

0/0

0/1

0/2

0/3

0/4

1/0

1/1

1/2

1/3

1/4

2/0

2/1

2/2

2/3

2/4

3/0

3/1

3/2

3/3

3/4

4/0

4/1

4/2

4/3

4/4

.

.

.

. . .

. . .

0/1

1/1

1/2

1/3

1/4

2/1

2/3

3/1

3/2

3/4

4/1

4/3

.

.

.

. . .

. . .

0

0/1

1/1

1/2

1/3

1/4

1

3

4

11

2

5

2/1

2/3

7

6

10

3/1

3/2

3/4

8

9

4/1

4/3

.

.

.

. . .

- The set Z of integers is countable

- The set Z of integers is countable
The following function f: Z ⟶ N is 1-1 and onto

…

-4

-3

-1

0

1

2

3

4

…

-2

- The set Z of integers is countable
The following function f: Z ⟶ N is 1-1 and onto

…

-4

7

-3

5

-1

1

0

0

2

1

2

4

6

3

4

8

…

-2

3

- Every Turing Machine can be given a unique number in binary as follows:
- Give to the states a number: q0 is 1, q1 is 11 etc…
- Give to the symbols of the tape a unique number: a is 1, b is 11, c is 111 etc…
- Assign 1 to L, 11 to R and 111 to S
- For each δ(q, a) = (q’, b, H) give the binary number 11…1011...1011…1011…1011…1
q a q’ b H

where H is L, R or S.

- Every Turing Machine can be given a unique number in binary as follows:
- To obtain the number of the machine combine the number of the transitions together (separated with 00).
- The number starts with q0 00 qf 00

- The number associated with the Turing Machine M is denoted as <M>

- The Turing Machine MP that computes the Predicate
n = 0

has the following number

<MP> = 100110010101101011

?

qf

□ → □, R

q0

- Countable sets are infinite.
- However there are some sets that are considered “even larger”.
- There is no way to enumerate them.

- Suppose that there is an enumeration of all the elements of the uncountable set.
- Obtain a new element by changing the ith place of the ith element.
- The new element is different than any other in the list.
- The new element is not in the enumeration.
Contradiction!!!

- Suppose that it was countable. An enumeration would be the following:
n0 =0.534103…

n1 =0.920531…

n2 =0.423327…

n3 =0.832739…

n4 =0.392098…

…

- Create a new number n by adding 1 mod 10 to the ith decimal digit of the (i-1)th number:
n0 =0.534103…

n1 =0.920531…

n2 =0.423327…

n3 =0.832739…

n4 =0.392098…

…

n =0.63480…

- The number n is different than any other number in the list.
n0 =0.534103…≠ 0.63480…

n1 =0.920531…≠ 0.63480…

n2 =0.423327…≠ 0.63480…

n3 =0.832739…≠ 0.63480…

n4 =0.392098…≠ 0.63480…

…

n =0.63480…

- Since n is a number in [0,1) it should be in the enumeration.
- Suppose that n = ni.
- This can’t be true because n and ni differ in the (i+1)th decimal digit.
Contradiction!

There is no possible enumeration for the numbers in [0,1), which means that:

The set of numbers in [0,1) is uncountable.

The set of real numbers contains the set [0,1). Thus it is uncountable.

Suppose that it was. An enumeration of all the predicate would be the following:

p0

p1

p2

p3

…

Suppose that it was. An enumeration of all the predicate would be the following:

pi(0)pi(1)pi(2)pi(3)…

p0 0 0 1 0

p10 1 0 0

p2 1 0 1 0

p30 1 1 1

… … … … …

Construct a predicate p as follows: Change the 0 with 1 and vice versa in pi(i)[ p(i) = 1-pi(i) ]

pi(0)pi(1)pi(2)pi(3)…

p00 0 1 0…

p10 1 0 0…

p2 1 0 1 0…

p30 1 1 1…

… … … … …

p 1 0 0 0…

Predicate p is not in the list since p(i) is different than pi(i) for every i.

pi(0)pi(1)pi(2)pi(3)…

p00 0 1 0…

p10 1 0 0…

p2 1 0 1 0…

p30 1 1 1…

… … … … …

p 1 0 0 0…

- Is the set of functions from N to {0,1,2} countable? Why?

- Is the set of functions from N to {0,1,2} countable? Why?
No. The set of functions from N to {0,1,2} contains the set of predicates.

Can you prove this by using diagonalization?

- Turing Machines are countable
- Predicates are uncountable
Thus there is a predicate for which there is no Turing Machine that decides it.