# Countability

• Nov 28th 2008, 10:09 AM
jas_viru
Countability
Q:
Let A,B≠0 and let f:A→B be an onto mapping. Then if A is countable then prove B is countable.
• Nov 28th 2008, 10:49 AM
Plato
Quote:

Originally Posted by jas_viru
Q:
Let A,B≠0 and let f:A→B be an onto mapping. Then if A is countable then prove B is countable.

This is one of the most important theorems in theory of cardinality of sets: $f: A \mapsto B \text{ is onto if and only if there is a one-to-one } g: B\mapsto A$.
Now any subset of a countable set is countable. Because $g(B)\subset A,\; g(B)\text { is countable }$

That means that $B \text{ is countable }$.