Slide 1Cantor’s Legacy: Infinity And DiagonalizationSlide 3An Ideal ComputerPrinting Out An Infinite SequenceComputable Real NumbersDescribable NumbersSlide 8Computable DescribableSlide 10Correspondence PrincipleCorrespondence DefinitionGeorg Cantor (1845-1918)Cantor’s Definition (1874)Do N and E have the same cardinality?Slide 16Slide 17Slide 18Do N and Z have the same cardinality?Slide 20Transitivity LemmaDo N and Q have the same cardinality?Slide 23Theorem: N and N×N have the same cardinalitySlide 25Slide 26Slide 27Slide 28Countable SetsDo N and R have the same cardinality?Theorem: The set R[0,1] of reals between 0 and 1 is not countableSlide 32Slide 33Slide 34Slide 35Diagonalized!Slide 37Slide 38Back to the questions we were asking earlierSlide 40Standard NotationTheorem: Every infinite subset S of S* is countableStringing Symbols TogetherSlide 44Slide 45Slide 46Slide 47Definition: Power SetSlide 49Slide 50Slide 51Slide 52Slide 53Slide 54Slide 55COMPSCI 102Introduction to Discrete MathematicsCantor’s Legacy: Infinity And DiagonalizationLecture 24 (November 18, 2009)The Theoretical Computer:no bound on amount of memoryno bound on amount of timeIdeal Computer is defined as a computer with infinite RAMYou can run a Java program and never have any overflow, or out of memory errorsAn Ideal ComputerIt can be programmed to print out:2: 2.0000000000000000000000…1/3: 0.33333333333333333333…: 1.6180339887498948482045…e: 2.7182818284559045235336…: 3.14159265358979323846264…Printing Out An Infinite SequenceA program P prints out the infinite sequence s0, s1, s2, …, sk, … if when P is executed on an ideal computer, it outputs a sequence of symbols such that- The kth symbol that it outputs is sk- For every k, P eventually outputs the kth symbol. I.e., the delay between symbol k and symbol k+1 is not infiniteComputable Real NumbersA real number R is computable if there is a (finite) program that prints out the decimal representation of R from left to right. Thus, each digit of R will eventually be output.Are all real numbers computable?Describable NumbersA real number R is describable if it can be denoted unambiguously by a finite piece of English text2: “Two.”: “The area of a circle of radius one.”Are all real numbers describable?Is every computable real number, also a describable real number?And what about the other way?Computable r: some program outputs rDescribable r: some sentence denotes rComputable DescribableTheorem: Every computable real is also describableProof: Let R be a computable real that is output by a program P. The following is an unambiguous description of R:“The real number output by the following program:” PAre all reals describable?Are all reals computable?We saw that computable describable, but do we also have describable computable?Correspondence PrincipleIf two finite sets can be placed into bijection, then they have the same sizeCorrespondence DefinitionIn fact, we can use the correspondence as the definition: Two finite sets are defined to have the same size if and only if they can be placed into bijectionGeorg Cantor (1845-1918)Cantor’s Definition (1874)Two sets are defined to have the same size if and only if they can be placed into bijectionTwo sets are defined to have the same cardinality if and only if they can be placed into bijectionDo and have the same cardinality?= { 0, 1, 2, 3, 4, 5, 6, 7, … } = { 0, 2, 4, 6, 8, 10, 12, … }The even, natural numbers.How can E and N have the same cardinality! E is a proper subset of N with plenty left over. The attempted correspondence f(x)=x does not take E onto N.E and N do have the same cardinality!N = 0, 1, 2, 3, 4, 5, … E = 0, 2, 4, 6, 8,10, …f(x) = x / 2 is a bijection(mapping E to N)Lesson: Cantor’s definition only requires that some injective correspondence between the two sets is a bijection, not that all injective correspondences are bijectionsThis distinction never arises when the sets are finiteDo and have the same cardinality?N = { 0, 1, 2, 3, 4, 5, 6, 7, … }Z = { …, -2, -1, 0, 1, 2, 3, … } and do have the samecardinality! = 0, 1, 2, 3, 4, 5, 6 … = 0, 1, -1, 2, -2, 3, -3, ….f(x) = x/2 if x is odd -x/2 if x is evenTransitivity LemmaHence, N, E, and Z all have the same cardinality.Lemma: If f: AB is a bijection, and g: BC is a bijection.Then h(x) = g(f(x)) defines a functionh: AC that is a bijectionDo N and Q have the same cardinality?N = { 0, 1, 2, 3, 4, 5, 6, 7, …. }Q = The Rational NumbersHow could it be????The rationals are dense: between any two there is a third. You can’t list them one by one without leaving out an infinite number of themTheorem: N and N×N have the same cardinality0 1 2 3 4 ……43210The point (x,y)represents the ordered pair (x,y)Onto the Rationals!The point at x,y represents x/yThe point at x,y represents x/y320 1Cantor’s 1877 letter to Dedekind:“I see it, but I don't believe it! ”Countable SetsWe call a set countable if it can be placed into a bijection with the natural numbers NHence N, E, Z, Q are all countableDo N and R have the same cardinality?N = { 0, 1, 2, 3, 4, 5, 6, 7, … }R = The Real NumbersTheorem: The set R[0,1] of reals between 0 and 1 is not countableProof: (by contradiction)Suppose R[0,1] is countableLet f be a bijection from N to R[0,1]Make a list L as follows:0: decimal expansion of f(0)1: decimal expansion of f(1) …k: decimal expansion of f(k)L0 1 2 3 4 …0123…IndexPosition after decimal pointL0 1 2 3 4 …03 3 3 3 3 313 1 4 1 5 921 2 4 8 1 234 1 2 2 6 8…IndexPosition after decimal pointL0 1 2 3 4 …0d01d12d23d3…d4L0 1 2 3 40d01d12d23d3……Define the following real numberConfuseL = 0.C0C1C2C3C4C5 …5, if dk=66, otherwiseCk=By design, ConfuseL can’t be on the list L! ConfuseL differs from the kth element on the list L in the kth position. This contradicts the assumption that the list L is complete; i.e., that the mapf: N to R[0,1] is onto.Diagonalized!The set of reals is uncountable!(Even the reals between 0 and 1)Why can’t the same argument be used to show that the set of rationals Q is uncountable?Since CONFUSEL is not necessarily rational, so there is no contradiction from the fact that it is missing from the list LBack to the questions we were asking earlierAre all reals describable?Are all reals computable?We saw that computable describable, but do we also have
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