1. © by Kenneth H. Rosen, Discrete Mathematics & its Applications, Sifth Edition, Mc Graw-Hill, 2007 Chapter 2: Basic Structures: Sets, Functions, Sequences and Sums Sets (Section 2.1) Sets (Section 2.1) Set Operations (Section 2.2) Set Operations (Section 2.2) Functions (Section 2.3) Functions (Section 2.3) Sequences and Summations (Section 2.4) Sequences and Summations (Section 2.4)

2. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 1 Sets (2.1) A set is a collection or group of objects or elements or members. (Cantor 1895) A set is a collection or group of objects or elements or members. (Cantor 1895) A set is said to contain its elements. A set is said to contain its elements. There must be an underlying universal set U, either specifically stated or understood. There must be an underlying universal set U, either specifically stated or understood.

3. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 2 Sets (2.1) (cont.) Notation: Notation: list the elements between braces: list the elements between braces: S = {a, b, c, d}={b, c, a, d, d} (Note: listing an object more than once does not change the set. Ordering means nothing.) specification by predicates: specification by predicates: S= {x| P(x)}, S contains all the elements from U which make the predicate P true. brace notation with ellipses: brace notation with ellipses: S = {..., -3, -2, -1}, the negative integers.

4. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 3 Sets (2.1) (cont.) Common Universal Sets Common Universal Sets R = reals R = reals N = natural numbers = {0,1, 2, 3,... }, the counting numbers N = natural numbers = {0,1, 2, 3,... }, the counting numbers Z = all integers = {.., -3, -2, -1, 0, 1, 2, 3, 4,..} Z = all integers = {.., -3, -2, -1, 0, 1, 2, 3, 4,..} Z + is the set of positive integers Z + is the set of positive integers Notation: Notation: x is a member of S or x is an element of S: x  S. x is not an element of S: x  S.

5. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 4 Sets (2.1) (cont.) Subsets Subsets Definition: The set A is a subset of the set B, denoted A  B, iff Definition: The set A is a subset of the set B, denoted A  B, iff  x [x  A  x  B] Definition: The void set, the null set, the empty set, denoted , is the set with no members. Definition: The void set, the null set, the empty set, denoted , is the set with no members. Note: the assertion x   is always false. Hence  x [x    x  B] is always true(vacuously). Therefore,  is a subset of every set. Note: A set B is always a subset of itself.

6. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 5 Sets (2.1) (cont.) Definition: If A  B but A  B the we say A is a proper subset of B, denoted A  B (in some texts). Definition: If A  B but A  B the we say A is a proper subset of B, denoted A  B (in some texts). Definition: The set of all subset of a set A, denoted P(A), is called the power set of A. Definition: The set of all subset of a set A, denoted P(A), is called the power set of A. Example: If A = {a, b} then Example: If A = {a, b} then P(A) = { , {a}, {b}, {a,b}}

7. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 6 Sets (2.1) (cont.) Definition: The number of (distinct) elements in A, denoted |A|, is called the cardinality of A. Definition: The number of (distinct) elements in A, denoted |A|, is called the cardinality of A. If the cardinality is a natural number (in N), then the set is called finite, else infinite. Example: A = {a, b}, Example: A = {a, b}, |{a, b}| = 2, |P({a, b})| = 4. A is finite and so is P(A). Useful Fact: |A|=n implies |P(A)| = 2 n

8. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 7 Sets (2.1) (cont.) N is infinite since |N| is not a natural number. It is called a transfinite cardinal number. N is infinite since |N| is not a natural number. It is called a transfinite cardinal number. Note: Sets can be both members and subsets of other sets. Note: Sets can be both members and subsets of other sets. Example: Example: A = { ,{  }}. A has two elements and hence four subsets: , {  }, {{  }}. { ,{  }} Note that  is both a member of A and a subset of A! Russell's paradox: Let S be the set of all sets which are not members of themselves. Is S a member of itself? Russell's paradox: Let S be the set of all sets which are not members of themselves. Is S a member of itself? Another paradox: Henry is a barber who shaves all people who do not shave themselves. Does Henry shave himself? Another paradox: Henry is a barber who shaves all people who do not shave themselves. Does Henry shave himself?

9. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 8 Sets (2.1) (cont.) Definition: The Cartesian product of A with B, denoted Definition: The Cartesian product of A with B, denoted A x B, is the set of ordered pairs { | a  A  b  B} Notation: Note: The Cartesian product of anything with  is . (why?) Example: Example: A = {a,b}, B = {1, 2, 3} AxB = {,,,,, } What is BxA? AxBxA? If |A| = m and |B| = n, what is |AxB|? If |A| = m and |B| = n, what is |AxB|?

10. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 9 Set Operations (2.2) (cont.) Propositional calculus and set theory are both instances of an algebraic system called a Propositional calculus and set theory are both instances of an algebraic system called a Boolean Algebra. The operators in set theory are defined in terms of the corresponding operator in propositional calculus As always there must be a universe U. All sets are assumed to be subsets of U

11. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 10 Set Operations (2.2) (cont.) Definition: Two sets A and B are equal, denoted A = B, iff Definition: Two sets A and B are equal, denoted A = B, iff  x [x  A  x  B]. Note: By a previous logical equivalence we have Note: By a previous logical equivalence we have A = B iff  x [(x  A  x  B)  (x  B  x  A)] or A = B iff A  B and B  A

12. 11 Set Operations (2.2) (cont.) Definitions: Definitions: The union of A and B, denoted A U B, is the set {x | x  A  x  B} The union of A and B, denoted A U B, is the set {x | x  A  x  B} The intersection of A and B, denoted A  B, is the set The intersection of A and B, denoted A  B, is the set {x | x  A  x  B} Note: If the intersection is void, A and B are said to be disjoint. The complement of A, denoted, is the set {x |  (x  A)} The complement of A, denoted, is the set {x |  (x  A)} Note: Alternative notation is A c, and {x|x  A}. The difference of A and B, or the complement of B relative to A, denoted A - B, is the set A  The difference of A and B, or the complement of B relative to A, denoted A - B, is the set A  Note: The (absolute) complement of A is U - A. The symmetric difference of A and B, denoted A  B, is the set The symmetric difference of A and B, denoted A  B, is the set (A - B) U (B - A)

13. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 12 Set Operations (2.2) (cont.) Examples: U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Examples: U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A= {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}. Then A  B = {1, 2, 3, 4, 5, 6, 7, 8} A  B = {1, 2, 3, 4, 5, 6, 7, 8} A  B = {4, 5} A  B = {4, 5} = {0, 6, 7, 8, 9, 10} = {0, 6, 7, 8, 9, 10} = {0, 1, 2, 3, 9, 10} = {0, 1, 2, 3, 9, 10} A - B = {1, 2, 3} A - B = {1, 2, 3} B - A = {6, 7, 8} B - A = {6, 7, 8} A  B = {1, 2, 3, 6, 7, 8} A  B = {1, 2, 3, 6, 7, 8}

14. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 13 Set Operations (2.2) (cont.) Venn Diagrams Venn Diagrams A useful geometric visualization tool (for 3 or less sets) A useful geometric visualization tool (for 3 or less sets) The Universe U is the rectangular box The Universe U is the rectangular box Each set is represented by a circle and its interior Each set is represented by a circle and its interior All possible combinations of the sets must be represented All possible combinations of the sets must be represented Shade the appropriate region to represent the given set operation. Shade the appropriate region to represent the given set operation. UU A A B C B For 2 sets For 3 sets

15. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 14 Set Operations (2.2) (cont.) Set Identities Set Identities Set identities correspond to the logical equivalences. Set identities correspond to the logical equivalences. Example: Example: The complement of the union is the intersection of the complements: =  =  Proof: To show:  x [x   x   ] To show two sets are equal we show for all x that x is a member of one set if and only if it is a member of the other.

16. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 15 Set Operations (2.2) (cont.) We now apply an important rule of inference (defined later) called We now apply an important rule of inference (defined later) called Universal Instantiation In a proof we can eliminate the universal quantifier which binds a variable if we do not assume anything about the variable other than it is an arbitrary member of the Universe. We can then treat the resulting predicate as a proposition.

17. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 16 We say We say 'Let x be arbitrary.' Then we can treat the predicates as propositions:

18. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 17 Set Operations (2.2) (cont.) Hence x   x   is a tautology. Since x was arbitrary x was arbitrary we have used only logically equivalent assertions and definitions we have used only logically equivalent assertions and definitions

19. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 18 Set Operations (2.2) (cont.) we can apply another rule of inference called Universal Generalization We can apply a universal quantifier to bind a variable if we have shown the predicate to be true for all values of the variable in the Universe. and claim the assertion is true for all x, i.e.,  x [x   x   ] Q. E. D. (Latin phrase “ Quod Erat Demonstrandum ” )

20. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 19 Set Operations (2.2) (cont.) Note: As an alternative which might be easier in some cases, use the identity Note: As an alternative which might be easier in some cases, use the identity A = B  [A  B and B  A] Example: Example: Show A  (B - A) =  The void set is a subset of every set. Hence, A  (B - A)   Therefore, it suffices to show A (B - A)  or  x [x  A (B - A)  x  ] A  (B - A)   or  x [x  A  (B - A)  x   ] So as before we say 'let x be arbitrary ’.

21. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 20 Set Operations (2.2) (cont.) Example (cont.) Example (cont.) Show x  A  (B - A)  x   is a tautology. But the consequent is always false. Therefore, the antecedent better always be false also. Apply the definitions:

22. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 21 Set Operations (2.2) (cont.) Example (cont.) Example (cont.) Hence, because P   P is always false, the implication is a tautology. The result follows by Universal Generalization. Q. E. D.

23. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 22 Set Operations (2.2) (cont.) Union and Intersection of Indexed Collections Union and Intersection of Indexed Collections Let A 1,A 2,..., A n be an indexed collection of sets. Let A 1,A 2,..., A n be an indexed collection of sets. Union and intersection are associative (because 'and' and 'or' are) we have: Union and intersection are associative (because 'and' and 'or' are) we have:

24. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 23 Set Operations (2.2) (cont.) Examples Let Examples Let

25. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 24 Functions (2.3) Definition: Let A and B be sets. A function (mapping, map) f from A to B, denoted f :A  B, is a subset of A*B such that Definition: Let A and B be sets. A function (mapping, map) f from A to B, denoted f :A  B, is a subset of A*B such that  x [x  A   y [y  B   f ]] and [  f   f ]  y 1 = y 2

26. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 25 Functions (2.3) (cont.) Note: f associates with each x in A one and only one y in B. Note: f associates with each x in A one and only one y in B. A is called the domain and B is called the codomain. If f(x) = y y is called the image of x under f x is called a preimage of y (note there may be more than one preimage of y but there is only one image of x). The range of f is the set of all images of points in A under f. We denote it by f(A).

27. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 26 Functions (2.3) (cont.) If S is a subset of A then f(S) = {f(s) | s in S}. Example: f(a) = Z f(a) = Z the image of d is Z the image of d is Z the domain of f is A = {a, b, c, d} the domain of f is A = {a, b, c, d} the codomain is B = {X, Y, Z} the codomain is B = {X, Y, Z} f(A) = {Y, Z} f(A) = {Y, Z} the preimage of Y is b the preimage of Y is b the preimages of Z are a, c and d the preimages of Z are a, c and d f({c,d}) = {Z} f({c,d}) = {Z} AB a b c d X Y Z

28. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 27 Functions (2.3) (cont.) Injections, Surjections and Bijections Injections, Surjections and Bijections Let f be a function from A to B. Let f be a function from A to B. Definition: f is one-to-one (denoted 1-1) or injective if preimages are unique. Definition: f is one-to-one (denoted 1-1) or injective if preimages are unique. Note: this means that if a  b then f(a)  f(b). Definition: f is onto or surjective if every y in B has a preimage. Definition: f is onto or surjective if every y in B has a preimage. Note: this means that for every y in B there must be an x in A such that f(x) = y. Definition: f is bijective if it is surjective and injective (one-to-one and onto). Definition: f is bijective if it is surjective and injective (one-to-one and onto).

29. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 28 Functions (2.3) (cont.) Examples: Examples: The previous Example function is neither an injection nor a surjection. Hence it is not a bijection. AB a b c d X Y Z Surjection but not an injection

30. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 29 Functions (2.3) (cont.) AB a b c d X Y Z W V AB a b c d X Y W V Injection but not a surjection Injection & a surjection, hence a bijection

31. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 30 Functions (2.3) (cont.) Note: Whenever there is a bijection from A to B, the two sets must have the same number of elements Note: Whenever there is a bijection from A to B, the two sets must have the same number of elements or the same cardinality. That will become our definition, especially for infinite sets. That will become our definition, especially for infinite sets.

32. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 31 Functions (2.3) (cont.) Examples: Examples: Let A = B = R, the reals. Determine which are injections, surjections, bijections: f(x) = x, f(x) = x, f(x) = x 2, f(x) = x 2, f(x) = x 3, f(x) = x 3, f(x) = x + sin(x), f(x) = x + sin(x), f(x) = | x | f(x) = | x |

33. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 32 Functions (2.3) (cont.) Let E be the set of even integers {0, 2, 4, 6,....}. Let E be the set of even integers {0, 2, 4, 6,....}. Then there is a bijection f from N to E, the even nonnegative integers, defined by f(x) = 2x. Hence, the set of even integers has the same cardinality as the set of natural numbers. OH, NO! IT CAN ’ T BE....E IS ONLY HALF AS BIG!!! Sorry! It gets worse before it gets better.

34. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 33 Functions (2.3) (cont.) Inverse Functions Inverse Functions Definition: Let f be a bijection from A to B. Then the inverse of f, denoted f -1, is the function from B to A defined as Definition: Let f be a bijection from A to B. Then the inverse of f, denoted f -1, is the function from B to A defined as f -1 (y) = x iff f(x) = y

35. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 34 Functions (2.3) (cont.) Example: Let f be defined by the diagram: Example: Let f be defined by the diagram: AB a b c d X Y W V f AB a b c d X Y W V f -1 Note: No inverse exists unless f is a bijection

36. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 35 Functions (2.3) (cont.) Definition: Let S be a subset of B. Then Definition: Let S be a subset of B. Then f-1(S) = {x | f(x)  S} Note: f need not be a bijection for this definition to hold. Example: Let f be the following function: Example: Let f be the following function: f -1 ({Z}) = {c, d} f -1 ({X, Y}) = {a, b} AB a b c d X Y Z

37. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 36 Functions (2.3) (cont.) Composition Composition Definition: Let f: B  C, g: A  B. The composition of f with g, denoted f  g, is the function from A to C defined by Definition: Let f: B  C, g: A  B. The composition of f with g, denoted f  g, is the function from A to C defined by f  g(x) = f(g(x)) f  g(x) = f(g(x))

38. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 37 Examples: Examples: AB a b c d X Y W V g j i h Cf A a b c d j i h C fgfg

39. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 38 Functions (2.3) (cont.) If f(x) = x 2 and g(x) = 2x + 1, then f(g(x)) = (2x+1) 2 and g(f(x)) = 2x 2 + 1 If f(x) = x 2 and g(x) = 2x + 1, then f(g(x)) = (2x+1) 2 and g(f(x)) = 2x 2 + 1 Definition: Definition: The floor function, denoted f ( x) =  x  or f(x) = floor(x), is the largest integer less than or equal to x. The floor function, denoted f ( x) =  x  or f(x) = floor(x), is the largest integer less than or equal to x. The ceiling function, denoted f ( x) =  x  or f(x) = ceiling(x), is the smallest integer greater than or equal to x. The ceiling function, denoted f ( x) =  x  or f(x) = ceiling(x), is the smallest integer greater than or equal to x.

40. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 39 Functions (2.3) (cont.) Examples:  3.5  = 3,  3.5  = 4. Examples:  3.5  = 3,  3.5  = 4. Note: the floor function is equivalent to truncation for positive numbers. Example: Example: Suppose f: B  C, g: A  B and f  g is injective. What can we say about f and g? We know that if a  b then f(g(a))  f(g(b)) since the composition is injective. We know that if a  b then f(g(a))  f(g(b)) since the composition is injective. Since f is a function, it cannot be the case that g(a) = g(b) since then f would have two different images for the same point. Since f is a function, it cannot be the case that g(a) = g(b) since then f would have two different images for the same point. Hence, g(a)  g(b) Hence, g(a)  g(b) It follows that g must be an injection. However, f need not be an injection (you show).

41. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 40 Sequences and Summations (2.4) Definition: A sequence is a function from a subset of the natural numbers (usually of the form {0, 1, 2,... } to a set S. Definition: A sequence is a function from a subset of the natural numbers (usually of the form {0, 1, 2,... } to a set S. Note: the sets {0, 1, 2, 3,..., k} and {1, 2, 3, 4,..., k} are called initial segments of N. Notation: if f is a function from {0, 1, 2,...} to S we usually denote f(i) by a i and we write where k is the upper limit (usually  ).

42. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 41 Sequences and Summations (2.4) (cont.) Examples: Using zero-origin indexing, if f(i) = 1/(i + 1). then the Sequence f = {1, 1/'2,1/3,1/4,... } = {a 0, a 1, a 2, a 3,.. } Using one-origin indexing the sequence f becomes {1/2, 1/3,...} = {a 1, a 2, a 3,...}

43. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 42 Sequences and Summations (2.4) (cont.) Summation Notation Summation Notation Given a sequence we can add together a subset of the sequence by using the summation and function notation or more generally

44. 43 Sequences and Summations (2.4) (cont.) Examples: Similarity for the product notation:

45. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 44 Sequences and Summations (2.4) (cont.) Definition: A geometric progression is a sequence of the form a, ar, ar 2, ar 3, ar 4,.... Your book has a proof that (you can figure out what it is if r = 1). You should also be able to determine the sum if the index starts at k vs. 0 if the index starts at k vs. 0 if the index ends at something other than n (e.g., n-1, n+1, etc.). if the index ends at something other than n (e.g., n-1, n+1, etc.).

46. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 45 Sequences and Summations (2.4) (cont.) Cardinality Cardinality Definition: The cardinality of a set A is equal to the cardinality of a set B, denoted Definition: The cardinality of a set A is equal to the cardinality of a set B, denoted | A | = | B |, if there exists a bijection from A to B.

47. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 46 Sequences and Summations (2.4) Definition: If a set has the same cardinality as a subset of the natural numbers N, then the set is called countable. Definition: If a set has the same cardinality as a subset of the natural numbers N, then the set is called countable. If |A| = |N|, the set A is countably infinite. The (transfinite) cardinal number of the set N is aleph null =  0. If a set is not countable we say it is uncountable.

48. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 47 Sequences and Summations (2.4) Examples: Examples: The following sets are uncountable (we show later) The real numbers in [0, 1] The real numbers in [0, 1] P(N), the power set of N P(N), the power set of N Note: With infinite sets proper subsets can have the same cardinality. This cannot happen with finite sets. Note: With infinite sets proper subsets can have the same cardinality. This cannot happen with finite sets. Countability carries with it the implication that there is a listing of the elements of the set.

49. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 48 Sequences and Summations (2.4) Definition: | A |  | B | if there is an injection from A to B. Definition: | A |  | B | if there is an injection from A to B. Note: as you would hope, Theorem: Theorem: If | A |  | B | and | B |  | A | then | A | = | B |. This implies if there is an injection from A to B if there is an injection from A to B if there is an injection from B to A if there is an injection from B to A then then there must be a bijection from A to B there must be a bijection from A to B This is difficult to prove but is an example of demonstrating existence without construction. This is difficult to prove but is an example of demonstrating existence without construction. It is often easier to build the injections and then conclude the bijection exists. It is often easier to build the injections and then conclude the bijection exists.

50. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 49 Sequences and Summations (2.4) Example: Example: Theorem: If A is a subset of B then | A |  | B |. Proof: the function f(x) = x is an injection from A to B. Example: {0, 2, 5}|   0 Example: {0, 2, 5}|   0 The injection f: {0, 2, 5}  N defined by f(x) = x is shown below: 0 1 2 3 4 5 6 … 0 2 5

51. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 50 Sequences and Summations (2.4) Some Countably Infinite Sets Some Countably Infinite Sets The set of even integers E ( 0 is considered even) is countably infinite. Note that E is a proper subset of N, The set of even integers E ( 0 is considered even) is countably infinite. Note that E is a proper subset of N, Proof: Let f(x) = 2x. Then f is a bijection from N to E Z +, the set of positive integers is countably infinite. Z +, the set of positive integers is countably infinite. 0 1 2 3 4 5 6 … 0 2 4 6 8 10 12 …

52. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 51 Sequences and Summations (2.4) The set of positive rational numbers Q + is countably infinite. The set of positive rational numbers Q + is countably infinite. Proof: Z + is a subset of Q + so |Z + | =  0  |Q + |. Now we have to show that |Q + |   0. To do this we show that the positive rational numbers with repetitions, Q R, is countably infinite. Then, since Q + is a subset of Q R, it follows that |Q + |   0 and hence |Q + | =  0.

53. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 52 Sequences and Summations (2.4)

54. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 53 Sequences and Summations (2.4) The position on the path (listing) indicates the image of the bijective function f from N to Q R : The position on the path (listing) indicates the image of the bijective function f from N to Q R : f(0) = 1/1, f(1) = 1/2, f(2) = 2/1, f(3) = 3/1, and so forth. Every rational number appears on the list at least once, some many times (repetitions). Hence, |N| = |Q R | =  0. Q. E. D The set of all rational numbers Q, positive and negative, is countably infinite. The set of all rational numbers Q, positive and negative, is countably infinite.

55. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 54 Sequences and Summations (2.4) The set of (finite length) strings S over a finite alphabet A is countably infinite. The set of (finite length) strings S over a finite alphabet A is countably infinite. To show this we assume that A is nonvoid A is nonvoid There is an “ alphabetical ” ordering of the symbols in A There is an “ alphabetical ” ordering of the symbols in A Proof: List the strings in lexicographic order: all the strings of zero length, all the strings of zero length, then all the strings of length 1 in alphabetical order, then all the strings of length 1 in alphabetical order, then all the strings of length 2 in alphabetical order, then all the strings of length 2 in alphabetical order, etc. etc. This implies a bijection from N to the list of strings and hence it is a countably infinite set.

56. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 55 Sequences and Summations (2.4) For example: For example: Let A = {a, b, c}. Then the lexicographic ordering of A is {, a, b, c, aa, ab, ac, ba, bb, bc, ca, cb, cc, aaa, aab, aac, aba,....} = {f(0), f(1), f(2), f(3), f(4),....}

57. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 56 Sequences and Summations (2.4) The set of all C programs is countable. The set of all C programs is countable. Proof: Let S be the set of legitimate characters which can appear in a C program. A C compiler will determine if an input program is a syntactically correct C program (the program doesn't have to do anything useful). A C compiler will determine if an input program is a syntactically correct C program (the program doesn't have to do anything useful). Use the lexicographic ordering of S and feed the strings into the compiler. Use the lexicographic ordering of S and feed the strings into the compiler. –If the compiler says YES, this is a syntactically correct C program, we add the program to the list. –Else we move on to the next string. In this way we construct a list or an implied bijection from N to the set of C programs. In this way we construct a list or an implied bijection from N to the set of C programs. Hence, the set of C programs is countable. Q. E. D. Hence, the set of C programs is countable. Q. E. D.

58. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 57 Sequences and Summations (2.4) Cantor Diagonalization Cantor Diagonalization An important technique used to construct an object which is not a member of a countable set of objects with (possibly) infinite descriptions An important technique used to construct an object which is not a member of a countable set of objects with (possibly) infinite descriptions Theorem: The set of real numbers between 0 and 1 is uncountable. Proof: We assume that it is countable and derive a contradiction. If it is countable we can list them (i.e., there is a bijection from a subset of N to the set). We show that no matter what list you produce we can construct a real number between 0 and 1 which is not in the list. Hence, there cannot exist a list and therefore the set is not countable

59. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 58 Sequences and Summations (2.4) It's actually much bigger than countable. It is said to have the cardinality of the continuum, c. Represent each real number in the list using its decimal expansion. e.g., 1/3 =.3333333........ 1/2 =.5000000........ 1/2 =.5000000........ =.4999999........ =.4999999........ If there is more than one expansion for a number, it doesn't matter as long as our construction takes this into account.

60. 59 Sequences and Summations (2.4) THE LIST.... THE LIST.... r 1 =.d 11 d 12 d 13 d 14 d 15 d 16..... r 2 =.d 21 d 22 d 23 d 24 d 25 d 26.... r 3 =.d 31 d 32 d 33 d 34 d 35 d 36....... Now construct the number x =.x 1 x 2 x 3 x 4 x 5 x 6 x 7.... x i = 3 if d ii  3 x i = 4 if d ii = 3 (Note: choosing 0 and 9 is not a good idea because of the non uniqueness of decimal expansions.) Then x is not equal to any number in the list. Hence, no such list can exist and hence the interval (0,1) is uncountable. Q. E. D.

61. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 60 Sequences and Summations (2.4) An extra goody: An extra goody: Definition: a number x between 0 and 1 is computable if there is a C program which when given the input i, will produce the ith digit in the decimal expansion of x. Example: The number 1/3 is computable. Example: The number 1/3 is computable. The C program which always outputs the digit 3, regardless if the input, computes the number.

62. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions 61 Sequences and Summations (2.4) Theorem: There is exists a number x between 0 and 1 which is not computable. There does not exist a C program (or a program in any other language) which will compute it! Why? Because there are more numbers between 0 and 1 than there are C programs to compute them. (in fact there are c such numbers!) Our second example of the nonexistence of programs to compute things!