1. ## Proof: Cardinality Question

If S is an uncountable set and T is a countable subset of S. Show that cardinality of |S\T|= |S|.

S\T is a subset of S, so |S\T|<= |S|.
I was going to use the cantor-schroder-bernstein theorem, but i dont know how to proof that |S\T|=> |S|.

If there is other ways to proof this, please post it.
Any help would be nice.

2. You can prove this by contradiction: if both $S\setminus T$ and $T$ are countable, then $S\subseteq (S\setminus T)\cup T$ is countable.

3. Originally Posted by firebio
If S is an uncountable set and T is a countable subset of S. Show that cardinality of |S\T|= |S|.

S\T is a subset of S, so |S\T|<= |S|.
I was going to use the cantor-schroder-bernstein theorem, but i dont know how to proof that |S\T|=> |S|.

If there is other ways to proof this, please post it.
Any help would be nice.
You can prove this by contradiction: if both $S\setminus T$ and $T$ are countable, then $S\subseteq (S\setminus T)\cup T$ is countable.
Haven't you only technically shown that $S-T$ is uncountable? But there are more than one cardinal number bigger than $\aleph_0$...
Wikipedia says that $\kappa_1+\kappa_2=\max(\kappa_1,\kappa_2)$ for infinite cardinals $\kappa_1, \kappa_2$ using the axiom of choice, but I don't know off top of my hat either how to prove this or if there is an easier proof.