Chapter 1. The Real Numbers 34 Exercise 1.6.4. Let S be the set consisting of al

Question

Chapter 1. The Real Numbers 34 Exercise 1.6.4. Let S be the set consisting of all sequences of 0's and 1's Observe that S is not a particular sequence, but rather a large set whose ele- ments are sequences; namely, S = {(a, a2, a3, . . .) : an-0 or 1} As an example, the sequence (1,0,1,0,1,0, 1,0,...) is an element of S, as is the sequence (1, 1,1, 1,1,1,...). Give a rigorous argument showing that S is uncountable. Having distinguished between the countable infinity of N and the uncount- able infinity of R, a new question that occupied Cantor was whether or not there existed an infinity "above" that of R. This is logically treacherous territory The same care we gave to defining the relationship "has the same cardinality as" needs to be given to defining relationships such as "has cardinality greater than" or "has cardinality too weighed down with formal definitions, one gets a very clear sense from the next result that there is a hierarchy of infinite sets that continues well beyond the continuum of R less than or equal to." Nevertheless, without getting

Explanation / Answer

Consider a sequence in S

We show its correspondence with a subset of N,set of natural numbers.

If, ai=0 then it means i is not present in the subset

If, ai=1 means i is present in the subset

THis gives us a one to one correspondence between a sequence in S and a subset of N.

Hence, S is in one to one correspondence with power set of N

But power set of N has cardinality , 2^N=|R|, cardinality of real numbers

Hence, S is uncountable

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