I'm sorry I still don't get it.
I don't understand what (c) and (d) are and how do their elements look like.
If you need to visualize them to help you see the problem better, try visualizing them in terms of the suggestion made by DrSteve earlier in this thread.
(c) Functions from to are essentially infinite sequences of and – e.g. etc. Can you visualize an infinite list of such infinite sequences?
(d) Functions from to can be regarded as sets of two ordered pairs of the form where are natural numbers. Can you visualize an infinite list of such sets?
To answer your question about irrational, algebraic and transcendental numbers:
An algebraic number is the solution of a polynomial equation with integer coefficients. For example, the fraction m/n is a solution of nx-m=0. This shows that all rational numbers are algebraic numbers. But there are also lots of algebraic numbers that are irrational. For example, the polynomial equation x^2-c=0 gives two algebraic numbers that are irrational whenever c is not a perfect square. It is not hard to show that the set of algebraic numbers is countable. Therefore, there are uncountably many transcendental numbers.
Thanks DrSteve, and thank you all for your time and your patience,
I believe that the source of confusion came along with something I read somewhere, where it said that "all the Transcendental numbers are not Algebraic" therefore I concluded that all the Irrationals should be Transcendental however, now I understand my mistake since gives two irrational numbers that are not transcendental since they solve the last polynomial equation.
by the way I figured out (c) and (d):
(c) .
This should generate a set with the following sequences:
(d) .
This should generate a set which contains sets of the form:
Is that correct?
Specifying a countable subset of some set X is more easily done by giving a sequence of elements of X (i.e., a function from to X) than by using a set-builder notation {x ∈ X | ...}. So a sequence of function in can be defined by
A sequence of function in can be defined by