Welcome to CS103! Two Handouts Today: Course Overview Introduction to Set Theory The Limits of Computation

Transcription

1 Welcome to CS103! Two Handouts Today: Course Overview Introduction to Set Theory The Limits of Computation

2 Course Staff Keith Schwarz Kyle Brogle Maurizio Calo Caligaris Julius Cheng Justin Heermann Nicholas Isaacs Michael Kim Tarun Singh Remington Wong Course Staff Mailing List:

3 The Course Website

4 Prerequisite CS106A

5 Prerequisite CS106A

6 Recommended Reading

7 Recommended Reading (but just the first chapter)

8 Online Course Notes

9 Grading Policies 60% Assignments 15% Midterm 25% Final

10 Goals for this Course

11 Key Questions How do we prove something with absolute certainty? Discrete Mathematics Can computers solve every problem? Computability Theory Why are some problems harder than others? Complexity Theory

12 Introduction to Set Theory

13 CS103 students All the computers on the Stanford network. Cool people The chemical elements Cute animals US coins.

14 ,,, Set Set notation: notation: Curly Curly braces braces with with commas commas separating separating out out the the elements elements A set is an unordered collection of distinct objects, which may be anything (including other sets).

15 ,,, These These are are the the same same set! set!,,, A set is an unordered collection of distinct objects, which may be anything (including other sets).

16 , These are are the the same same set! set!,,,,, A set is an unordered collection of distinct objects, which may be anything (including other sets).

17 This symbol means is defined as Ø We denote it with this symbol The empty set contains no elements. A set is an unordered collection of distinct objects, which may be anything (including other sets).

18 Membership,,, Is in this set?

19 Membership,,, Is in this set?

20 Set Membership Given a set S and an object x, we write x S if x is contained in S, and otherwise. x S If x S, we say that x is an element of S. Given any object and any set, either that object is an element of the set or it isn't.

21 Infinite Sets Sets can be infinitely large. The natural numbers, N: { 0, 1, 2, 3, } Some authors don't include zero; in this class, assume that 0 is a natural number. The integers, Z: {, -2, -1, 0, 1, 2, } Z is from German Zahlen. The real numbers, R, including rational and irrational numbers.

22 Constructing Sets from Other Sets Consider these English descriptions: All even numbers. All real numbers less than 137. All negative integers. We can't list their (infinitely many!) elements. How would we rigorously describe them?

23 The Set of Even Numbers { x x N and x is even } The set of all x where x is in the set of natural numbers and x is even

24 Set Builder Notation A set may be specified in set-builder notation: { x some property x satisfies } For example: { r r R and r < 137 } { n n is a power of two } { x x is a set of US currency } { p p is a legal Java program }

25 Combining Sets

26 Venn Diagrams A B A = { 1, 2, 3 } B = { 3, 4, 5 }

27 Venn Diagrams A A B A = { 1, 2, 3 } B = { 3, 4, 5 }

28 Venn Diagrams B A B A = { 1, 2, 3 } B = { 3, 4, 5 }

29 Venn Diagrams Union A B { 1, 2, 3, 4, 5 } A B A = { 1, 2, 3 } B = { 3, 4, 5 }

30 Venn Diagrams Intersection A B { 3 } A B A = { 1, 2, 3 } B = { 3, 4, 5 }

31 Venn Diagrams Difference A B { 1, 2 } A B A = { 1, 2, 3 } B = { 3, 4, 5 }

32 Venn Diagrams Difference A \ B { 1, 2 } A B A = { 1, 2, 3 } B = { 3, 4, 5 }

33 Venn Diagrams Symmetric Difference A Δ B { 1, 2, 4, 5 } A B A = { 1, 2, 3 } B = { 3, 4, 5 }

34 Venn Diagrams A B A Δ B

35 Venn Diagrams

36 Venn Diagrams for Three Sets

37 Venn Diagrams for Four Sets B C A D Question Question to to ponder: ponder: why why can't can't we we just just draw draw four four circles? circles?

38 Venn Diagrams for Five Sets

39 A Fun Website: Venn Diagrams for Seven Sets

40 Subsets and Power Sets

41 Subsets A set S is a subset of a set T (denoted S T) if all elements of S are also elements of T. Examples: { 1, 2, 3 } { 1, 2, 3, 4 } N Z (every natural number is an integer) Z R (every integer is a real number)

42 What About the Empty Set? A set S is a subset of a set T (denoted S T) if all elements of S are also elements of T. Is Ø S for any set S? Yes: The above statement is always true. Vacuous truth: A statement that is true because it does not apply to anything. All unicorns are blue. All unicorns are pink.

43 Proper Subsets By definition, any set is a subset of itself. (Why?) A proper subset of a set S is a set T such that T S T S There are multiple notations for this; they all mean the same thing: T S T S

44 S =, (S) = Ø,,,, (S) (S) is is the the power power set set of of S (the (the set set of of all all subsets subsets of of S) S)

45 Cardinalities

46 Cardinality The cardinality of a set is the number of elements it contains. We denote it S. Examples: { a, b, c, d, e} = 5 { {a, b}, {c, d, e, f, g}, {h} } = 3 { 1, 2, 3, 3, 3, 3, 3 } = 3 { x x N and x < 137 } = 137

47 The Cardinality of N What is N? There are infinitely many natural numbers. N can't be a natural number, since it's infinitely large. We need to introduce a new term. Definition: N = ℵ 0 Pronounced Aleph-Zero, Aleph-Nought, or Aleph-Null.

48 Consider the set S = { x x N and x is even } What is S?

49 How Big Are These Sets?,,,,,,

50 Comparing Cardinalities Two sets have the same cardinality if their elements can be put into a one-to-one correspondence with one another. The intuition:,,,,,,

51 Comparing Cardinalities Two sets have the same cardinality if their elements can be put into a one-to-one correspondence with one another. The intuition:,,,,, We've run out of elements in the second set!

52 Infinite Cardinalities N S n 2n S = { x x N and x is even } S = N = ℵ 0

53 Infinite Cardinalities N Z N = Z = ℵ 0 n n / 2 (if n is even) n -(n + 1) / 2 (if n is odd)

54 Infinite Cardinalities N P n 2 n P = { x x N and x is a power of two} P = N = ℵ 0

55 Important Question Do all infinite sets have the same cardinality?

56 Prepare for one of the most beautiful (and surprising!) proofs in mathematics...

57 S =, (S) = Ø,,,, S < (S)

58 S =,, Ø,,,, (S) =,,,,,,,, S < (S)

59 S = {a, b, c, d} (S) = { Ø, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {b, e} {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, {a, b, c, d} } S < ( S)

60 If S is infinite, what is the relation between S and ( S)? Does S = ( S)?

61 If S = ( S), there has to be a one-to-one correspondence between elements of S and subsets of S. What might this correspondence look like?

62 x 0 x 1 x 2 x 3 x 4 x 5... x 0 x 1 x 2 x 3 x 4 x 5 { x 0, x{ 2, 0, x 4 2,,... 4, }... } { x 0, x{ 3, 0, x 4 2,,... 4, }... } { x 4,... { 0, } 2, 4,... } { x 1, x{ 4, 0,... 2, } 4,... } { b, x 0, x{ 5 ab,, 0,... 2, } 4, } } { x 0, x{ 1, 0, x 2 2,, x4, 3, x... 4, } x 5,... }...

63 x 0 x 1 x 2 x 3 x 4 x 5... x 0 x 1 x 2 Y N{ 0, Y2, 4, N... } Y N Y N N Y Y N N N N N Y N x 3 N Y N N Y N x 4 x 5... { N Y b, N{ ab, 0, NY 2, 4,... NY... } N} Y N Y Y Y Y Y Y Y

64 x 0 x 1 x 2 x 3 x 4 x 5... x 0 x 1 x 2 Y N{ 0, Y2, 4, N... } Y N Y N N Y Y N N N N N Y N x 3 N Y N N Y N x 34 x { N Y b, N{ ab, 0, NY 2, 4,... NY... } N} Y N Y Y Y Y Y Y Y

65 x 0 x 1 x 2 x 3 x 4 x 5... x 0 x 1 x 2 x 3 x 34 x 45 Y N{ 0, Y2, 4, N... } Y N Y N N Y Y N N N N N Y N N Y N N Y N { N Y b, N{ ab, 0, NY 2, 4,... NY... } N} Y N Y Y Y Y Y Y Y Which Which row row in in the the table table is is paired paired with with this this set? set?... Y N N N N Y

66 x 0 x 1 x 2 x 3 x 4 x 5... x 0 x 1 x 2 x 3 x 34 x 45 Y N{ 0, Y2, 4, N... } Y N Y N N Y Y N N N N N Y N N Y N N Y N { N Y b, N{ ab, 0, NY 2, 4,... NY... } N} Y N Y Y Y Y Y Y Y Flip Flip all all Y's Y's to to N's N's and and vice-versa vice-versa to to get get a a new new set set... NY NY NY NY NY N Y...

67 x 0 x 1 x 2 x 3 x 4 x 5... x 0 x 1 x 2 Y N{ 0, Y2, 4, N... } Y N Y N N Y Y N N N N N Y N x 3 x 34 N N Y Y N N Y N N NY NY NY N Y Which Which row row in in the the table table is is paired paired with with this this set? set? x Y Y Y Y Y Y NY NY NY NY N Y N Y...

68 x 0 x 1 x 2 x 3 x 4 x 5... x 0 x 1 x 2 Y N{ 0, Y2, 4, N... } Y N Y N N Y Y N N N N N Y N x 3 x 34 N N Y Y N N Y N N NY NY NY N Y Which Which row row in in the the table table is is paired paired with with this this set? set? x Y Y Y Y Y Y NY NY NY NY N Y N Y...

69 x 0 x 1 x 2 x 3 x 4 x 5... x 0 x 1 x 2 Y N{ 0, Y2, 4, N... } Y N Y N N Y Y N N N N N Y N x 3 x 34 N N Y Y N N Y N N NY NY NY N Y Which Which row row in in the the table table is is paired paired with with this this set? set? x Y Y Y Y Y Y NY NY NY NY N Y N Y...

70 x 0 x 1 x 2 x 3 x 4 x 5... x 0 x 1 x 2 Y N{ 0, Y2, 4, N... } Y N Y N N Y Y N N N N N Y N x 3 x 34 N N Y Y N N Y N N NY NY NY N Y Which Which row row in in the the table table is is paired paired with with this this set? set? x Y Y Y Y Y Y NY NY NY NY N Y N Y...

71 x 0 x 1 x 2 x 3 x 4 x 5... x 0 x 1 x 2 Y N{ 0, Y2, 4, N... } Y N Y N N Y Y N N N N N Y N x 3 x 34 N N Y Y N N Y N N NY NY NY N Y Which Which row row in in the the table table is is paired paired with with this this set? set? x Y Y Y Y Y Y NY NY NY NY N Y N Y...

72 x 0 x 1 x 2 x 3 x 4 x 5... x 0 x 1 x 2 Y N{ 0, Y2, 4, N... } Y N Y N N Y Y N N N N N Y N x 3 x 34 N N Y Y N N Y N N NY NY NY N Y Which Which row row in in the the table table is is paired paired with with this this set? set? x Y Y Y Y Y Y NY NY NY NY Y N Y...

73 x 0 x 1 x 2 x 3 x 4 x 5... x 0 x 1 x 2 Y N{ 0, Y2, 4, N... } Y N Y N N Y Y N N N N N Y N x 3 x 34 N N Y Y N N Y N N NY NY NY N Y Which Which row row in in the the table table is is paired paired with with this this set? set? x Y Y Y Y Y Y NY NY NY NY N Y N Y...

74 x 0 x 1 x 2 x 3 x 4 x 5... x 0 x 1 x 2 Y N{ 0, Y2, 4, N... } Y N Y N N Y Y N N N N N Y N x 3 x 34 N N Y Y N N Y N N NY NY NY N Y Which Which row row in in the the table table is is paired paired with with this this set? set? x Y Y Y Y Y Y NY NY NY NY N Y N Y...

75 The Diagonalization Proof The complemented diagonal cannot appear anywhere in the table. In row n, the nth element must be wrong. No matter how we try to assign subsets of S to elements of S, there will always be at least one subset left over. Cantor's Theorem: Every set is smaller than its power set: For any set S, S < ( S)

76 Infinite Cardinalities Recall: N = ℵ₀. By Cantor's Theorem: N < (N) (N) < ( (N)) ( (N)) < ( ( (N))) ( ( (N))) < ( ( ( (N)))) Not all infinite sets have the same size. There are many different infinities.

77 What does this have to do with computation?

78 The set of all computer programs The set of all problems to solve

79 Strings and Problems Consider the set of all strings: {, a, b, c,..., aa, ab, ac, } For any set of strings S, we can solve the following problem about S: Write a program that accepts as input a string, then prints out whether or not that string belongs to set S. Therefore, there are at least as many problems to solve as there are sets of strings.

80 Every computer program is a string. So, there can't be any more programs than there are strings. From Cantor's Theorem, we know that there are more sets of strings than strings. There are at least as many problems as there are sets of strings. < Programs Strings Sets of Strings Problems

81 Every computer program is a string. So, there can't be any more programs than there are strings. From Cantor's Theorem, we know that there are more sets of strings than strings. There are at least as many problems as there are sets of strings. Programs < Problems

82 There are more problems to solve than there are programs to solve them.

83 It Gets Worse Because there are more problems than strings, we can't even describe some of the problems that we can't solve. Using more advanced set theory, we can show that there are infinitely more problems than solutions. In fact, if you pick a totally random problem, the probability that you can solve it is zero.

84 But then it gets better...

85 Where We're Going Given this hard theoretical limit, what can we compute? What are the hardest problems we can solve? How powerful of a computer do we need to solve these problems? Of what we can compute, what can we compute efficiently? What tools do we need to reason about this? How do we build mathematical models of computation? How can we reason about these models?

86 Next Time Mathematical Proof What is a mathematical proof? How can we prove things with certainty?