# Union of countable sets is countable

## Homework Statement

Prove that a finite union of countable sets is also countable. Is an infinite union of countable sets also countable?

## Homework Equations

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

## 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:

tiny-tim
Homework Helper
Hi BiP! Then each element of {A+B} has an injective mapping to N, since each element of {A+B} is in A or in B.

Sorry, but that doesn't make any sense Why don't you try to construct a mapping? haruspex