Is the cardinality of an infinitely countable set the same as the rational numbers?

Answer 1

Yes.

There is an injective function from rational numbers to positive rational numbers*.

Every positive rational number can be written in lowest terms as a/b, so there is an injective function from positive rationals to pairs of positive integers.

The function f(a,b) = a^2 + 2ab + b^2 + a + 3b maps maps every pair of positive integers (a,b) to a unique integer.

So there is an injective function from rationals to integers.

Since every integer is rational, the identity function is an injective function from integers to rationals.

Then By the Cantor-Schroder-Bernstein theorem, there is a bijective function from rationals to integers, so the rationals are countably infinite.

*This is left as an exercise for the reader.