Number Theory (Finite and Infinite Sets)

[Ankit Mishra]

Homework Statement

Why is R\Q not countably infinite or denumerable? Given R (Real Number) is not countably infinite or denumerable and Q (rational number) is denumerable.

Homework Equations

A set is said to be denumberable or countably infinite if there exists a bijestion of N (natural Number) onto S.

The Attempt at a Solution

Let Q be a subset of R and Let S be R-Q, which is denumerable. Via the defn there exists a bijection but R-Q is not bijective R-Q is the set of Real Numbers which is already not denumerable. I showed this to my prof and he said its not correct. He said to use Q u (R-Q)=R, we know Q is denumerable and R not to be denumerable use this to show R-Q is not denumerable? HOW!

 

[clamtrox]

What you need to show is that the union of two countably infinite sets is countable. So take two sets {A_n} and {B_n}. Can you find a bijection from their union to the natural numbers? (hint: it's really easy)