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Kenneth Rosen Edition 7 Exercise 2.5 Question 37 (Page No. 177)
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Show that the set of all computer programs in a particular programming language is countable. [Hint: A computer program written in a programming language can be thought of as a string of symbols from a finite alphabet.]
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Kenneth Rosen Edition 7 Exercise 2.5 Question 40 (Page No. 177)
Show that if $S$ is a set, then there does not exist an onto function $f$ from $S$ to $P(S),$ the power set of $S$. Conclude that $\mid S\mid < \mid P(S)\mid .$ This result is known as Cantor's theorem. [Hint: Suppose such a function ... $s$ can exist for which $f (s) = T.]$
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Kenneth Rosen Edition 7 Exercise 2.5 Question 39 (Page No. 177)
We say that a function is computable if there is a computer program that finds the values of this function. Use question $37$ and $38$ to show that there are functions that are not computable.
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Kenneth Rosen Edition 7 Exercise 2.5 Question 38 (Page No. 177)
Show that the set of functions from the positive integers to the set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$ is uncountable. [Hint: First set up a one-to-one correspondence between the set of real numbers between $0$ and $1$ and a subset of ... to the real number $0.\:d_{1}d_{2} \dots d_{n}\dots $ the function $f$ with $f (n) = dn.]$
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Kenneth Rosen Edition 7 Exercise 2.5 Question 36 (Page No. 177)
Show that there is a one-to-one correspondence from the set of subsets of the positive integers to the set real numbers between $0$ and $1$. Use this result and question $34$ and $35$ to conclude that $ℵ_{0} < \mid P(Z^{+})\mid =\mid R\mid.\:[$Hint: Look at the first part of the hint for Exercise $35.]$
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