kenneth rosen Ex2.4 Q.46

Question

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.

Comments

@Arjun Sir, Please explain the proof, Despite reading from Rosen,I m not comfortable with this topic!

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@Bhaskar Singh it works. But "we all know that" is false. Even majority doesn't know :)

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@himgta two answers are given here. If you're not understanding then basically you have to revise countability, power set, infinite set etc. These are pretty simple but you shouldn't have even a minute misunderstanding here. Also this topic and most of discrete requires good brain work -- there's no other way. 

Answer 1

This is the explanation given online.But I m not able to understand as to why set of real numbers is mentioned.

we know that the set of real numbers between 0 and 1(
denoted by (0, 1) is uncountable. Let us associate to each real number [0, 1) a function from
the set of positive integers to the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} as follows:
If x is a real number whose decimal representation is 0.d 1 d 2 d 3 · · · (with ambiguity resolved
by forbidding the decimal to end with a infinite string of 9’s). The we associate to the x the
function whose rule is given by f (n) = d n .
Clearly, this is a one to one function from [0,1) and a subset of all functions from positive
integers to the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Two different real numbers must have different
decimal representation, so the corresponding functions are different.(A few functions are left
out, because of forbidding representations such as 0.2399999 · · ·).
Since (0,1) is uncountable, the subset of functions we have associated with them must be
uncountable. But the set of all such functions has at least this cardinality, so it, too, must