# Union of countable sets is countable

1. May 18, 2013

### Bipolarity

1. The problem statement, all variables and given/known data
Prove that a finite union of countable sets is also countable. Is an infinite union of countable sets also countable?

2. Relevant equations
A set S is countable if and only if there exists an injection from S to N.

3. The attempt at a solution
I will attempt prove it for the case of 2 sets. Proving it for a finite collection of sets follows analogously. Suppose the countable sets are A and B. Then there are injections $f_{A}$ and $f_{B}$ from A to N and B to N respectively. We need to show the existence of an injection from {A+B} to N where + denotes union.

Since {A+B} is the union of A and B, certainly it contains an element that is in at least A or in B (or in both A and B). Then each element of {A+B} has an injective mapping to N, since each element of {A+B} is in A or in B.

Does this complete the proof? Is this rigorous?

And what about the case for an infinite union?

BiP

Last edited: May 18, 2013
2. May 18, 2013

### tiny-tim

Hi BiP!
Sorry, but that doesn't make any sense

Why don't you try to construct a mapping?

3. May 19, 2013

### haruspex

You need a mapping that avoids mapping an element of A-B and an element of B-A to the same element in N.