a) Show that the union of a countable set of countable sets is countable.

(Remember, countable means that the set is either finite or countably infinite. You need to discuss the various cases that arise because of this option.)

Solved here: https://www.chegg.com/homework-help/questions-and-answers/show-union-countable-set-countable-sets-countable-remember-countable-means-set-either-fini-q27231983

b) Use the result from a) to write a more formal answer to: 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.]

c) 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 these functions. Do this by associating to the real number 0.d 1 d 2 ...d n ... the function f with f(n) = d n .]

[Hint 2: If r = 0.1357924680 ... then the function f they define in the hint is such that f (1) = 1, f (2) = 3, f (6) = 2, f (9) = 8 and so on..]