A Very Practical Series

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What if we also save a fixed amount (d) every year?

Formulate the Recurrence Relation•  

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Let r = interest rate, d = amount saved annually.Let k = 1+r.

Formulate the Recurrence Relation•  

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Let r = interest rate, d = amount saved annually.Let k = 1+r.

Infinite Sets and Cardinality•  

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Infinite Sets and Cardinality•  

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The Schroeder-Bernstein Theorem: If

then .

(We will use this result w/o proof.)

(Bijection)

Infinite Sets and Cardinality•  

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Apparent Paradoxes•  

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Apparent Paradoxes•  

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-4 -2 0 2 4

3 1 0 2 4

Apparent Paradoxes•  

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-4 -2 0 2 4

3 1 0 2 4

Clearly(?) one-to-one and onto.

Apparent Paradoxes•  

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Apparent Paradoxes• A proper subset of a finite set has a smaller

cardinality than the whole set.

• A proper subset of an infinite set can have the same cardinality as the whole set.– | Even_Integers | = | Z | = ℵ0 .

• Hilbert’s “Grand Hotel”.

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The Rational Numbers are Countable

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Consider positive rational

numbersfirst

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Question

• Are you convinced by this proof that the set of positive rationals, Q+, is countable?– (A) Yes! That’s really clever!– (B) Hmm . . . Why is this surprising?– (C) Hmm . . . What about duplicate values in

Grid?– (D) I want to see what to do with negative

rational numbers.– (E) I just don’t get it.

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What about negative rational numbers?• Just use a different Grid.

• The set Q is countable. 25

The Real Numbers are Not CountableProposition ¬C: R is not countable.

Proposition C: R is countable.• If R is countable, then (0,1) is countable.• Then there is a 1-1 correspondence between

Z+ and (0,1).• Let that correspondence be this table:

• Every ri (0,1) must be in this table.∈26

1 r1 = 0.d11 d12 d13 d14 . . .

2 r2 = 0.d21 d22 d23 d24 . . .

3 r3 = 0.d31 d32 d33 d34 . . .

4 r4 = 0.d41 d42 d43 d44 . . .

. . . . . .

The Real Numbers are Not CountableProposition ¬C: R is not countable.

Proposition C: R is countable.• If R is countable, then (0,1) is countable.• Then there is a 1-1 correspondence between

Z+ and (0,1).• Let that correspondence be this table:

• Every ri (0,1) must be in this table.∈27

1 r1 = 0.d11 d12 d13 d14 . . .

2 r2 = 0.d21 d22 d23 d24 . . .

3 r3 = 0.d31 d32 d33 d34 . . .

4 r4 = 0.d41 d42 d43 d44 . . .

. . . . . .

The Real Numbers are Not CountableProposition ¬C: R is not countable.

Proposition C: R is countable.• If R is countable, then (0,1) is countable.• Then there is a 1-1 correspondence between

Z+ and (0,1).• Let that correspondence be this table:

• Every ri (0,1) must be in this table.∈28

1 r1 = 0.d11 d12 d13 d14 . . .

2 r2 = 0.d21 d22 d23 d24 . . .

3 r3 = 0.d31 d32 d33 d34 . . .

4 r4 = 0.d41 d42 d43 d44 . . .

. . . . . .

The Real Numbers are Not CountableProposition ¬C: R is not countable.

Proposition C: R is countable.• If R is countable, then (0,1) is countable.• Then there is a 1-1 correspondence between

Z+ and (0,1).• Let that correspondence be this table:

• Every ri (0,1) must be in this table.∈29

1 r1 = 0.d11 d12 d13 d14 . . .

2 r2 = 0.d21 d22 d23 d24 . . .

3 r3 = 0.d31 d32 d33 d34 . . .

4 r4 = 0.d41 d42 d43 d44 . . .

. . . . . .

The Real Numbers are Not CountableProposition ¬C: R is not countable.

Proposition C: R is countable.• If R is countable, then (0,1) is countable.• Then there is a 1-1 correspondence between

Z+ and (0,1).• Let that correspondence be this table:

• Every ri (0,1) must be in this table.∈30

1 r1 = 0.d11 d12 d13 d14 . . .

2 r2 = 0.d21 d22 d23 d24 . . .

3 r3 = 0.d31 d32 d33 d34 . . .

4 r4 = 0.d41 d42 d43 d44 . . .

. . . . . .

The Real Numbers are Not CountableProposition ¬C: R is not countable.

Proposition C: R is countable.• If R is countable, then (0,1) is countable.• Then there is a 1-1 correspondence between

Z+ and (0,1).• Let that correspondence be this table:

• Every ri (0,1) must be in this table.∈31

1 r1 = 0.d11 d12 d13 d14 . . .

2 r2 = 0.d21 d22 d23 d24 . . .

3 r3 = 0.d31 d32 d33 d34 . . .

4 r4 = 0.d41 d42 d43 d44 . . .

. . . . . .

The Real Numbers are Not Countable

• Construct a number x = 0. x1 x2 x3 x4 . . . where each digit xi ≠ dii, for every i.– Therefore, x ≠ ri, for every i.

– So, x is not in the table. Even though x (0,1).∈

• Thus, Proposition C leads to a contradiction.– Therefore, Proposition ¬C is true:

• The set R is not countable. Q.E.D.

– This is a diagonalization proof by contradiction.32

The Real Numbers are Not Countable

• Construct a number x = 0. x1 x2 x3 x4 . . . where each digit xi ≠ dii, for every i.– Therefore, x ≠ ri, for every i.

– So, x is not in the table. Even though x (0,1).∈

• Thus, Proposition C leads to a contradiction.– Therefore, Proposition ¬C is true:

• The set R is not countable. Q.E.D.

– This is a diagonalization proof by contradiction.33

The Real Numbers are Not Countable

• Construct a number x = 0. x1 x2 x3 x4 . . . where each digit xi ≠ dii, for every i.– Therefore, x ≠ ri, for every i.

– So, x is not in the table. Even though x (0,1).∈

• Thus, Proposition C leads to a contradiction.– Therefore, Proposition ¬C is true:

• The set R is not countable. Q.E.D.

– This is a diagonalization proof by contradiction.34

Question

• Are you convinced by this proof, that the set R of real numbers is uncountable?

– (A) Yes! That’s really clever!– (B) Hmph. Can’t you just add x to the table?– (C) Hmph. Can you write down that table?– (D) Hmph. Why does getting a contradiction

from C mean that ¬C must be true?– (E) I just don’t get it.

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