Real analysis, function from I to I * I ,?

Question

a) Let I be the set of decimals of the form 0.d1d2......( numbers are subscripts) construct a one to one function from I to I x I ( I cross I).
b) find either an onto function from I to IxI or a one to one function for IxI to I.
c) do I and IxI have the same cardinality
help please,
source:
excercise 2.1.3 , pp34, intr to real analysis, Micheal Schramm

Answer 1

For part a:
f(x) = (x, x)

For part b:
Let d(x, n) denote the nth digit of the infinite decimal x. Now define f:I×I→I using:
d(f(x, y), n) = {d(x, n/2) if n is even, d(y, (n+1)/2) if n is odd}
This function is one-to-one and onto. In fact, you could have used the inverse of this function for part a, if you wanted to.

For part c: Yes, as there exists a bijection between them.

Edit: hey Steve, just saw your question. Actually, when I say decimals here, I am considering them as infinite sequences directly, and not as representations of real numbers - thus the decimals 0.1999999.... and 0.200000.... are distinct decimals, even though they represent the same real number. Since the problem made no mention of real numbers (aside from the name of the textbook), I assumed that was the proper context. You will note that I have been careful to refer to them only has decimals and not as decimal expansions, since we are not expanding anything here.

Answer 2

Just to be a nuisance, I'd like to ask Pascal to be more specific about what he means by decimal expansion. If he admits repeating 9's, then his function is not one to one, since both
(0.99999999..., 0.19999999999...) and (0,0.2) map to 0.2.

If he does not admit repeating 9's then it is not onto, as nothing can map to 0.1919191919...

Edit: Ah, you're not talking about real numbers. Well that's just confusing...

Answer 3

x0 is in Q, then we are able to discover a chain x_n completely in (no longer Q) converging to x0, and vice versa if x0 is in (no longer Q). it somewhat is genuine through fact the two Q (the rationals) and (no longer Q) (the irrationals) are dense interior the set of exact numbers. in the two case, lim f(x_n) does no longer --> to f(x0) even nonetheless x_n --> x0.

Answer 4

a: Every real number, except zero, has a unique reciprocal. Try that.