0 tooo than all rational There're more real numbers in Q numbers True False
1 Cardinality of sets _IDefI Let S be a set If there are exactly NEIN distinct we say S is afinitesd wity elements in S Notation 1st denotes cardinality of S 0 I 2,3 4 A n t N I ne 53 IAI A EI 5 101 0 I 19031 IDefI A set is said to be infinite if it is not finite f A Recall B A labelled balls B labelled bins IDefI Two sets A and B have the same cardinality wrut en 1131 if there is a bijection from A to B I Al Let S be the set of even integers Prove that Is I I 21 EI S Pf O 2 4 4 2 l l I l l 21 I I 2 O 2 Consider f S such that f n 21 2n To show f is injective Sta Stb a b Suppose f Ca fcb for some ai b C 21
Thus 2 a 2b a b Thus f is injective To show f is surjective t z s S Let s c S 1 Furthermore f Is 2 I S Then E S E Z S Thus f is surjective D or has the same cardinality as µ is IDefI A set that is finite called c k Red An infinite set S is countable if we can list elements in S because f IN in a sequence a o ai az S given by f n an is a bijection is countable E.g Z I 21 2 3 3 O I The set offinite length bitstrings is countable O l OO ol 10 11 000 001 010 I Al E lBl HHMI Schroder Bernstein If there exist injections 1131 E I Al and g between sets A and B then there exists a bijection h A B
7 Qt is countable go i z Obviously there's an injection from Nl to Qt Pf we need to find an injection from Qt to Nl Recall that Qt Ma l p g c It as I Ii F F do r r and I 432,42 k at's I 3 e I I I I is an injection from t.to N min n so q an _q I KINI D Reem It followsthat IQ is countable as well
I I Cantor diagonalization argument R is uncountable Assume IR is countable PI ra.se pyffxe Since EO I C Rl Eo IT is countable List elements in EO I as ro ri r z Let the decimal representation of them as O doo To do doz To O 00000 r O d o di d z ri O 1415926 R2 O 3261 O d w du der Tz i 0.100 f Form a real number with deciomal expansion 0 dod d z ith digit of ri r dii 1 if such that di O 0 diito if Then r differs at the ith digit with ri so ti r tri r is a real number not on our list 0,1 is not countable so R is not countable Hence B Red Similarly the set of infinite length bit strings is uncountable si TotunTount Red Be careful with uncountable sets lock CZ CQ CR LET I However Erik n o
2 Un computable Functions IDefI A function is computable if there is a computer program in some programming language that finds the value of this function IThmI There are an computable functions Chaim There're uncountably many functions from IX to N PI Suppose there're countably many functions from Nl to Nl PI 210 fall f 214 f fo 6 H fill 11 fly 11 f A N not on our list conclude B A computer program is a bit string with finite length 3 computer programs is countable there're are un computable functions b
u poPoPiP2 ring HH Y Pill uncomputable function 2 I an example L L L Pz Pgdm input D8 Tuiring Define Test Halt P if program P halts on input x x yes if program P loops on input x no IThmITest Hatt is uncomputable Assume TestHalt is computable PI aPCR halts Define loop forever if Test Hatt Pip Turing P yes Hatt Pip halt if Test no what is Turing Turing If Turing Turing halts Matt f Turing Turing Test no loops forever Turing Turing If Turing Turing loops forever D Remi A common strategy to show a program P is uncomputable using P to implement testhalt is i e reducing Test Matt to P Hatt computable Test P computable
IPropI A is countable Given B E A then B is countable The statement obviously holds it A or B is finite pf So assume A B ane infinite F bijection f A is countable A N Restrict f on BEA to get f B an injection N Then f i B SIBI is a bijection infinitesubset N of N is countable Cja An PILATE N recursively by eotinemingn X t.me Dg in aiiie.oi Since f B is an infinite sublet of N by the claim f B is countable i e there exists a bijection g HB Nl Thus go f i B i e B is countable N is a bijection B Is f B 9 i N
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