Aim: Infinite Sets Course: Math Lit. Do Now: Aim: How can the word ‘infinite’ define a collection of elements? In the PBS science program Cosmos: A Personal

Aim: Infinite Sets Course: Math Lit.

Do Now:

Aim: How can the word ‘infinite’ define a collection of elements?

In the PBS science program Cosmos: A Personal Voyage, Episode 9: "The Lives of the Stars", astronomer and television personality Carl Sagan estimated that writing a googolplex in numerals (i.e., "10,000,000,000...") would be physically impossible, since doing so would require more space than the known universe provides.

What is a googolplex?1001010


One-to-one Correspondence

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 0

6-1-7.5 2

Number and its place on a number line

Mapping relations

01234

-3-1135

-3-1135

relation A

01234

A

relation A–1

A–1


One-to-One Function

• Only one-to-one functions have inverses that are functions.

• A one-to-one function passes the horizontal line test i.e. the line crosses the function at one and only one point.

4

2

-2

-4

q x = x34

2

-2

-4

• A function f has an inverse function f-1 if and only if f is one-to-one.


One-to-One Correspondence

A one-to-one correspondence (1 – 1 correspondence) between two sets A and B is a rule or procedure that pairs each element of A with exactly one element of B and each element of B with exactly one element of A.Consider:

Concert hall has 890 seats

For one performance all seats are occupied

no need to count to know attendance

For another performance six seats are empty.

no need to count to know attendance: 890 – 6 = 884.


One-to-One Correspondence

Two sets A and B are equivalent, denoted by A ~ B, if and only if A and B can be placed in a

one-to-one correspondence.

Equivalent Sets – contain the same number of elements. Cardinalities are equal: n(A) = n(B) or |A| = |B|.

expanded

A = {a, b, c, d, e}

B = {1, 2, 3, 4, 5} A ~ B

n(A) = n(B) |A| = |B|

= 5

A = {a, b, c, d, e}

B = {1, 2, 3, 4, 5}


Model Problem

Establish a one-to-one correspondence between the set of natural numbers N = {1, 2, 3, 4, 5, . . . , n, . . .} and the set of even natural numbers E = {2, 4, 6, 8, 10, . . . , 2n, . . .}

N = {1, 2, 3, 4, 5, . . . , n, . . .}

E = {2, 4, 6, 8, 10, . . . , 2n, . . .}

n N

2n N

n 2ndetermines exactly what elements are paired in N & E

17 34 76867 153734

and establishes a one-to-one correspondence between sets

N ~ E


Definition of Infinite Set

A set is an infinite set if it can be placed in a one-to-one correspondence with a proper subset of itself. N ~ E

Can the set {1, 2, 3} be placed in a one-to- one correspondence with one of its proper subsets?

No, {1, 2, 3} is finite and every proper subset of {1, 2, 3} has 2 or fewer elements: {1}; {1, 2}; {2, 3}, etc.


Model Problem

Verify the S = {5, 10, 15, 20, . . . 5n, . . .} is an infinite set.

plan: establish a one-to-one correspondence with a proper subset

T = {10, 20, 30, 40, . . . 10n, . . .}

S = {5, 10, 15, 20, . . . 5n, . . .}

10n T

5n S

This general correspondence 5n 10n establishes a one-to-one with one of S’s proper subsets (T) making S an infinite set.


Model Problem

Verify the V = {40, 41, 42, 43, . . . 40 + n, . . .} is an infinite set.

plan: establish a one-to-one correspondence with a proper subset


Cardinality of Infinite Sets

o (read as aleph-null) represents the cardinal number for the set N of natural numbers. n(N) = o

Since o has a cardinality greater than any infinite number it is called a transfinite number.

Many infinite sets have a cardinality of o.

Does an infinite set have cardinality?

Georg Cantor

‘How can one infinity be greater than another?”


Model Problem

Show that the set of integers I = {. . . , -4, -3, -2, -1, 0 , 1, 2, 3, 4, . . .} has a cardinality of o.

I = {. . . , -4, -3, -2, -1, 0 , 1, 2, 3, 4, . . .}

N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 11, . . . }

no obvious one-to-one correspondence

I = {0, 1, -1, 2, -2, 3, -3, 4, . . . , –n + 1, n, . . .}

N = {1, 2, 3, 4, 5, 6, 7, 8, . . . , 2n – 1, 2n, . . . }however, with rearranging of I

each even natural number 2n of set N is paired with the integer n of set I. (blue arrows)each odd natural number 2n – 1 of N is paired with the integer –n + 1 of I. (red arrows)

(2n) n and (2n – 1) (-n + 1): 1-2-1

The set of integers has a cardinality of o.

general correspondence


Model Problem

0

1 1 1 1 1Show that { , , , , , , }

2 3 4 5 1has a cardinality of .

Mn


Theorem of Rational Numbers

The set Q+ of positive rational numbers is equivalent to the set N of natural numbers.

1 2 3 4 5 6 7

1 1 1 1 1 1 11 3 5 7 9 11 13

2 2 2 2 2 2 21 2 4 5 7 8 10

3 3 3 3 3 3 31 3 5 7 9 11 13

4 4 4 4 4 4 4

expressed in lowest terms; smallest to largest

Rational number in array

Corresponding natural number 1 2 3 4 5 6 7 8 9 10

1

1

2

1

1

2

4

1

5

2

2

3

1

4

Which rational is assigned to

N = 11?


Countable Set

A set is a countable set if and only if it is a finite set or an infinite set that is equivalent tothe set of natural numbers.

Every infinite set that is countable has a cardinality of o.

However, not all infinite sets are countable.

Ex: Theorem: The set of A = {x|x R and 0 < x < 1} is not a countable set.

Proof by contradiction – Assume it is a countable set and work until we arrive at a contradiction.


Proof by Contradiction

The set of A = {x|x R and 0 < x < 1} is not a countable set.assume countable

N = {1, 2, 3, 4, 5, 6, 7, 8, . . . n, . . . }

A = {a1, a2, a3, a4, a5, a6, a7, a8, . . . an, . . . }

1 a1 = 0.3573485 . . .

2 a2 = 0.0652891 . . .

3 a3 = 0.6823514 . . .

4 a4 = 0.0500310 . . .

n an = 0.3155728 . . . 5 . . .

0.4 7 3

build a number d using diagonal

method1 . . . 0 < < 1

contradiction!!! but it’s not is set A! It differs from each number of A in at least 1 decimal place

0 < d < 1


Countable/Uncountable

An infinite set that is not countable is saidto be uncountable. An uncountable set does not have a cardinality of o. It has a cardinality of c, from the word continuum.

c > o

Cardinality of Some Infinite Sets

Set Cardinal Number

Natural Numbers, N o

Integers, I o

Rational Numbers, Q o

Irrational Numbers c

Any set of form {x|a < x < b} a and b are real a b

c

Real Numbers, R c


Cantor’s Theorem

Cantor’s Theorem

Let S be any set. The set of all subsets of S has a cardinal number that is larger that the cardinal number of S.

This set is called the Power Set of S denoted by P(S)

recall: S = {1, 2, 3} |S| = 3

S has 23 = 8 subsets

Cantor said this is true for infinite sets as well

No matter how large the cardinal number of a setwe can find a set that has a larger cardinal number

There are infinitely many transfinite numbers


Transfinite Arithmetic Theorems

For any whole number a, o + a = o and o – a = o

o + o = o and in general, o + o + o + o = o

a finite number of o

c + c = c and in general, c + c + c + c = c

a finite number of c

c + o = c

c o = c


Documents

Aim: Infinite Sets Course: Math Lit. Do Now: Aim: How can the word ‘infinite’ define a collection of elements? In the PBS science program Cosmos: A Personal