18 Slides--Relations

18 Slides--Relations - CS103 HO#18 Slides-Relations April...

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CS103 HO#18 Slides--Relations April 16, 2010 1 Infinite Sets How about the rationals? Countably infinite—same size as N How about the reals? We will use a different kind of diagonal argument to show that the real numbers are not denumerable. In fact, we will show that we cannot count the reals in the open interval (0,1). R (0,1) = {x | (x R) (0 < x < 1)} Suppose we have an enumeration of R (0,1) , i.e., a list of all reals in the interval (0, 1). Z + R (0,1) 10 . d 11 d 12 d 13 ... 20 . d 21 d 22 d 23 ... 30 . d 31 d 32 d 33 ... ... i0 . d i1 d i2 d i3 ... ... where d ij {0, 1, 2, . ..,9} Suppose we have an enumeration of R (0,1) , i.e., a list of all reals in the interval (0, 1). Z + R (0,1) . d 11 d 12 d 13 ... . d 21 d 22 d 23 ... . d 31 d 32 d 33 ... ... . d i1 d i2 d i3 ... ... where d ij {0, 1, 2, . ..,9} Now we construct a new number x = 0.p 1 p 2 p 3 ... where pi = (d i + 1) mod 10 (we will not use 0 and 9 so that we do not produce duplicates). x is not in the list, since it differs from every number along the diagonal! We conclude that it is not possible to produce such a list. Infinite Sets How about the rationals? Countably infinite—same size as N | N | = 0 How about the reals? Uncountable—bigger than N Every real number in (0, 1) can be represented by an infinite fraction: 0.10000000. .. 0.17917917. .. 0.31415926. .. This is true regardless of the base of the number system used. For example: 0.1 (decimal) = 0.0 0011 0011 0011 0011 . . . (base 2) The fractional part has a 1 st , 2 nd , 3 rd , . .. digit
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CS103 HO#18 Slides--Relations April 16, 2010 2 Every real number in (0, 1) can be represented by an infinite fraction: 0.10000000. .. 0.17917917. .. 0.31415926. .. This is true regardless of the base of the number system used. For example: 0.1 (decimal) = 0.0 0011 0011 0011 0011 . . . (base 2) The fractional part has a 1 st , 2 nd , 3 rd , . .. digit So the digits of the fraction can be put in 1-to-1 correspondence with N So each fraction has | N | digits, i.e., 0 digits Every real number in (0, 1) can be represented by an infinite fraction: 0.10000000. .. 0.17917917. .. 0.31415926. .. This is true regardless of the base of the number system used. For example: 0.1 (decimal) = 0.0 0011 0011 0011 0011 . . . (base 2) The fractional part has a 1 st , 2 nd , 3 rd , . .. digit So the digits of the fraction can be put in 1-to-1 correspondence with N So each fraction has | N | digits, i.e., 0 digits We could form a subset of N by including the positions of the 1 digits For the above, this would be {4, 5, 8, 9, 12, 13, 16, 17, . ..} So every real in (0, 1) corresponds to a particular subset, and every subset corresponds to a particular real number Every real number in (0, 1) can be represented by an infinite fraction: 0.10000000. ..
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18 Slides--Relations - CS103 HO#18 Slides-Relations April...

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