1. A Very Practical Series 1 What if we also save a fixed amount (d) every year?

2. Formulate the Recurrence Relation 2 Let r = interest rate, d = amount saved annually. Let k = 1+r.

3. Formulate the Recurrence Relation 3 Let r = interest rate, d = amount saved annually. Let k = 1+r.

4. Infinite Sets and Cardinality 4

5. Infinite Sets and Cardinality 5 The Schroeder-Bernstein Theorem: If then. (We will use this result w/o proof.) (Bijection)

6. Infinite Sets and Cardinality 6

7. Apparent Paradoxes 7

8. Apparent Paradoxes 8 -4-2024 31024

9. Apparent Paradoxes 9 -4-2024 31024 Clearly(?) one-to-one and onto.

10. Apparent Paradoxes 10

11. 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 | = ℵ 0. Hilbert’s “Grand Hotel”. 11

12. The Rational Numbers are Countable 12 Consider positive rational numbers first

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24. 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. 24

25. What about negative rational numbers? Just use a different Grid. The set Q is countable. 25

26. The Real Numbers are Not Countable Proposition ¬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 r i ∈ (0,1) must be in this table. 26 1r 1 = 0.d 11 d 12 d 13 d 14... 2r 2 = 0.d 21 d 22 d 23 d 24... 3r 3 = 0.d 31 d 32 d 33 d 34... 4r 4 = 0.d 41 d 42 d 43 d 44......

27. The Real Numbers are Not Countable Proposition ¬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 r i ∈ (0,1) must be in this table. 27 1r 1 = 0.d 11 d 12 d 13 d 14... 2r 2 = 0.d 21 d 22 d 23 d 24... 3r 3 = 0.d 31 d 32 d 33 d 34... 4r 4 = 0.d 41 d 42 d 43 d 44......

28. The Real Numbers are Not Countable Proposition ¬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 r i ∈ (0,1) must be in this table. 28 1r 1 = 0.d 11 d 12 d 13 d 14... 2r 2 = 0.d 21 d 22 d 23 d 24... 3r 3 = 0.d 31 d 32 d 33 d 34... 4r 4 = 0.d 41 d 42 d 43 d 44......

29. The Real Numbers are Not Countable Proposition ¬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 r i ∈ (0,1) must be in this table. 29 1r 1 = 0.d 11 d 12 d 13 d 14... 2r 2 = 0.d 21 d 22 d 23 d 24... 3r 3 = 0.d 31 d 32 d 33 d 34... 4r 4 = 0.d 41 d 42 d 43 d 44......

30. The Real Numbers are Not Countable Proposition ¬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 r i ∈ (0,1) must be in this table. 30 1r 1 = 0.d 11 d 12 d 13 d 14... 2r 2 = 0.d 21 d 22 d 23 d 24... 3r 3 = 0.d 31 d 32 d 33 d 34... 4r 4 = 0.d 41 d 42 d 43 d 44......

31. The Real Numbers are Not Countable Proposition ¬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 r i ∈ (0,1) must be in this table. 31 1r 1 = 0.d 11 d 12 d 13 d 14... 2r 2 = 0.d 21 d 22 d 23 d 24... 3r 3 = 0.d 31 d 32 d 33 d 34... 4r 4 = 0.d 41 d 42 d 43 d 44......

32. The Real Numbers are Not Countable Construct a number x = 0. x 1 x 2 x 3 x 4... where each digit x i ≠ d ii, for every i. –Therefore, x ≠ r i, 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

33. The Real Numbers are Not Countable Construct a number x = 0. x 1 x 2 x 3 x 4... where each digit x i ≠ d ii, for every i. –Therefore, x ≠ r i, 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

34. The Real Numbers are Not Countable Construct a number x = 0. x 1 x 2 x 3 x 4... where each digit x i ≠ d ii, for every i. –Therefore, x ≠ r i, 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

35. 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. 35