# Is injectivity or surjectivity enough to show countability?

I'm new into Measure theory and I'm self-taught. For some time now, I have been encountering proofs that showed countability using injectivity or surjectivity. Here are examples https://proofwiki.org/wiki/Countable_Union_of_Countable_Sets_is_Countable/Proof_1 and https://proofwiki.org/wiki/Countable_Union_of_Countable_Sets_is_Countable/Proof_2

To me, a set is countable if and only if there exists a one-one correspondence or bijection between it and the set of natural numbers, $$\Bbb{N}.$$

MY QUESTION

Is injectivity or surjectivity enough to show countability as the above proofs have done?

• Your definition would refer to countably infinite – Hagen von Eitzen Sep 23 '18 at 16:27
• @Hagen von Eitzen: Countably infinite? Let me check! – Omojola Micheal Sep 23 '18 at 16:28
• I think both are required as indicated by wiki – poetasis Sep 23 '18 at 16:29
• From what I have gathered, a set is countable if and only if there exists a bijection between it and the set of natural numbers, $\Bbb{N}.$ From this source proofwiki.org/wiki/Definition:Countable_Set/Definition_1, injection is needed. From this one, proofwiki.org/wiki/Definition:Countable_Set/Definition_2, a set is countable if and only if it is finite or countably infinite and from here proofwiki.org/wiki/Definition:Countable_Set/Definition_3; a set is countable if and only if there exists a bijection between it and a subset of $\Bbb{N}$. So, which one do I follow? – Omojola Micheal Sep 23 '18 at 16:35
• This question has set a confusion to me for some time now. I don't know how to explain it when the time comes. It's an honest question. So, I think it deserves to be closed! – Omojola Micheal Sep 23 '18 at 16:38

## 1 Answer

The usual definitions are that:

1. a set is finite if it is in bijection with $$\{1 , 2, \cdots, n\}$$ for some $$n$$
2. a set is countably infinite if it is in bijection with $$\mathbb{N}$$
3. a set is countable if it is finite or countably infinite

Then, the following conditions are equivalent for a set $$S$$:

1. $$S$$ is countable
2. There is a surjection $$\mathbb{N} \to S$$
3. There is an injection $$S \to \mathbb{N}$$