Infinity and Uncountability. How big is the set of reals or the set - PowerPoint PPT Presentation

Infinity and Uncountability. How big is the set of reals or the set of integers? Same size? Same number? ◮ Countable Make a function f : Circles → Squares. Infinite! f ( red circle ) = red square ◮ Countably infinite. Is one bigger or smaller? f ( blue circle ) = blue square ◮ Enumeration f ( circle with black border ) = square with black border One to one. Each circle mapped to different square. One to One: For all x , y ∈ D , x � = y = ⇒ f ( x ) � = f ( y ) . Onto. Each square mapped to from some circle . Onto: For all s ∈ R , ∃ c ∈ D , s = f ( c ) . Isomorphism principle: If there is f : D → R that is one to one and onto, then, | D | = | R | . Isomorphism principle. Combinatorial Proofs. Countable. The number of subsets of a set { a 1 ,..., a n } . ? Given a function, f : D → R . Equal to the number of binary n -bit strings. How to count? One to One: f : Subsets. → Strings. 0, 1, 2, 3, ... For all ∀ x , y ∈ D , x � = y = ⇒ f ( x ) � = f ( y ) . f ( x ) = ( g ( x , a 1 ) , g ( x , a 2 ) ,..., g ( x , a n )) or The Counting numbers. The natural numbers! N ∀ x , y ∈ D , f ( x ) = f ( y ) = ⇒ x = y . � 1 a ∈ x Definition: S is countable if there is a bijection between S and some Onto: For all y ∈ R , ∃ x ∈ D , y = f ( x ) . g ( x , a ) = 0 otherwise subset of N . f ( · ) is a bijection if it is one to one and onto. If the subset of N is finite, S has finite cardinality . Example: Isomorphism principle: S = { 1 , 2 , 3 , 4 , 5 } , x = { 1 , 3 , 4 } . If the subset of N is infinite, S is countably infinite . If there is a bijection f : D → R then | D | = | R | . f ( x ) = ( 1 , 0 , 1 , 1 , 0 ) . | P ( S ) | = |{ 0 , 1 } n | = 2 n .

Where’s 0? A bijection is a bijection. More large sets. Which is bigger? The positive integers, Z + , or the natural numbers, N . Natural numbers. 0 , 1 , 2 , 3 , .... Notice that there is a bijection between N and Z + as well. Positive integers. 1 , 2 , 3 , .... f ( n ) = n + 1 . 0 → 1 , 1 → 2 , ... E - Even natural numbers? Where’s 0? Bijection from A to B = ⇒ a bijection from B to A . f : N → E . More natural numbers! f ( n ) → 2 n . Consider f : Z + → N where f ( z ) = z − 1 . Onto: ∀ e ∈ E , f ( e / 2 ) = e . e / 2 is natural since e is even For any two z 1 � = z 2 = ⇒ z 1 − 1 � = z 2 − 1 = ⇒ f ( z 1 ) � = f ( z 2 ) . One-to-one: ∀ x , y ∈ N , x � = y = ⇒ 2 x � = 2 y . ≡ f ( x ) � = f ( y ) One to one! Evens are countably infinite. Inverse function! Evens are same size as all natural numbers. For any natural number n , Can prove equivalence either way. for z = n + 1 , f ( z ) = ( n + 1 ) − 1 = n . Onto! Bijection to or from natural numbers implies countably infinite. Bijection! | Z + | = | N | . But.. but where’s zero? “It comes from 1.” All integers? Listings.. Enumerability ≡ countability. What about Integers, Z ? � n / 2 if n even Define f : N → Z . f ( n ) = − ( n + 1 ) / 2 if n odd. Enumerating (listing) a set implies that it is countable. � n / 2 if n even f ( n ) = Another View: − ( n + 1 ) / 2 if n odd. “Output element of S ”, n f ( n ) “Output next element of S ” 0 0 One-to-one: For x � = y ... 1 − 1 if x is even and y is odd, Any element x of S has specific, finite position in list. 2 1 then f ( x ) is nonnegative and f ( y ) is negative = ⇒ f ( x ) � = f ( y ) Z = { 0 , 1 , − 1 , 2 , − 2 , ..... } 3 − 2 if x is even and y is even, Z = {{ 0 , 1 , 2 ,..., } and then {− 1 , − 2 ,... }} 4 2 then x / 2 � = y / 2 = ⇒ f ( x ) � = f ( y ) When do you get to − 1? at infinity? ... ... .... Need to be careful. Onto: For any z ∈ Z , Notice that: A listing “is” a bijection with a subset of natural numbers. if z ≥ 0, f ( 2 z ) = z and 2 z ∈ N . Function ≡ “Position in list.” if z < 0, f ( 2 | z |− 1 ) = z and 2 | z | + 1 ∈ N . If finite: bijection with { 0 ,..., | S |− 1 } Integers and naturals have same size! If infinite: bijection with N .

Countably infinite subsets. Enumeration example. Fractions? Enumerating a set implies countable. All binary strings. Can you enumerate the rational numbers in order? Corollary: Any subset T of a countable set S is countable. B = { 0 , 1 } ∗ . 0 ,..., 1 / 2 ,.. Enumerate T as follows: B = { ε , 0 , 1 , 00 , 01 , 10 , 11 , 000 , 001 , 010 , 011 ,... } . Get next element, x , of S , Where is 1 / 2 in list? ε is empty string. output only if x ∈ T . After 1 / 3, which is after 1 / 4, which is after 1 / 5... For any string, it appears at some position in the list. Implications: If n bits, it will appear before position 2 n + 1 . A thing about fractions: Z + is countable. any two fractions has another fraction between it. It is infinite. Should be careful here. There is a bijection with the natural numbers. Does this mean we can’t even get to “next” fraction? B = { ε ; , 0 , 00 , 000 , 0000 ,... } So it is countably infinite. Can’t list in “order”? Never get to 1. All countably infinite sets have the same cardinality. Pairs of natural numbers. Pairs of natural numbers. Rationals? Positive rational number. Enumerate in list: Lowest terms: a / b ( 0 , 0 ) , ( 1 , 0 ) , ( 0 , 1 ) , ( 2 , 0 ) , ( 1 , 1 ) , ( 0 , 2 ) ,...... · · · · · a , b ∈ N 3 with gcd ( a , b ) = 1 . Consider pairs of natural numbers: N × N · · · · · Infinite subset of N × N . 2 E.g.: ( 1 , 2 ) , ( 100 , 30 ) , etc. Countably infinite! For finite sets S 1 and S 2 , · · · · · 1 then S 1 × S 2 All rational numbers? has size | S 1 |×| S 2 | . Negative rationals are countable. (Same size as positive rationals.) · · · · · 0 So, is N × N countably infinite squared ??? Put all rational numbers in a list. 0 1 2 3 4 The pair ( a , b ) , is in first ( a + b + 1 )( a + b ) / 2 elements of list! First negative, then nonegative ??? No! (i.e., “triangle”). Repeatedly and alternatively take one from each list. Countably infinite. Interleave Streams in 61A The rationals are countably infinite! Same size as the natural numbers!!

Real numbers.. The reals. Diagonalization. If countable, there a listing, L contains all reals. For example 0: . 500000000 ... 1: . 785398162 ... 2: . 367879441 ... Are the set of reals countable? 3: . 632120558 ... Lets consider the reals [ 0 , 1 ] . 4: . 345212312 ... . . Each real has a decimal representation. . Is the set of real numbers the “same size” as integers? . 500000000 ... (1 / 2) Construct “diagonal” number: . 77677 ... . 785398162 ... π / 4 . 367879441 ... 1 / e Diagonal Number: Digit i is 7 if number i ’s, i th digit is not 7 and 6 otherwise. . 632120558 ... 1 − 1 / e . 345212312 ... Some real number Diagonal number for a list differs from every number in list! Diagonal number not in list. Diagonal number is real. Contradiction! Subset [ 0 , 1 ] is not countable!! All reals? Diagonalization: Summary Another diagonalization. The set of all subsets of N . Assume is countable. There is a listing, L , that contains all subsets of N . 1. Assume that a set S can be enumerated. Subset [ 0 , 1 ] is not countable!! Define a diagonal set, D : 2. Consider an arbitrary list of all the elements of S . If i th set in L does not contain i , i ∈ D . What about all reals? 3. Use the diagonal from the list to construct a new element t . otherwise i �∈ D . No. 4. Show that t is different from all elements in the list D is different from i th set in L for every i . Any subset of a countable set is countable. = ⇒ t is not in the list. = ⇒ D is not in the listing. If reals are countable then so must [ 0 , 1 ] . 5. Show that t is in S . D is a subset of N . 6. Contradiction. L does not contain all subsets of N . Contradiction. Theorem: The set of all subsets of N is not countable. (The “set of all subsets of N ” is the powerset of N .)

Cardinalities of uncountable sets? Summary. Cardinality of [ 0 , 1 ] smaller than all the reals? f : R + → [ 0 , 1 ] . ◮ Bijections to equate cardinality of infinite sets x + 1 � 0 ≤ x ≤ 1 / 2 2 f ( x ) = 1 x > 1 / 2 ◮ Countable (infinite) sets 4 x ◮ Uncountable sets One to one. x � = y If both in [ 0 , 1 / 2 ] , a shift = ⇒ f ( x ) � = f ( y ) . ◮ Diagonalization If neither in [ 0 , 1 / 2 ] a division = ⇒ f ( x ) � = f ( y ) . If one is in [ 0 , 1 / 2 ] and one isn’t, different ranges = ⇒ f ( x ) � = f ( y ) . Bijection! [ 0 , 1 ] is same cardinality as nonnegative reals!