Prove that real numbers and irrational numbers are numerically equivalent.

Question

How do I prove that set A and set A/C are numerically equivalent,where A is uncountable set and C is countably finite subset of A. This can be further used to show that real numbers and irrational numbers are numerically equivalent.
I am aware of the fact that,what are uncountable and countably infinite sets and also the meaning of numerically equivalent.I am not able to create the required bijection.

Answer 1

Choose a countable subset $D$ of the irrationals, say all the rational translates of $\pi$. Then form a 1-1 correspondence between $D$ and $D\cup C$ where $C$ are the rationals. Together with the identity on $\mathbb{R}\setminus(C\cup D)$ this provides the desired correspondence.