Question about enumerability [closed]

If a set is enumerable, is the following true:

Edit: My definition of enumerable is: There is a one-to-one correspondence between the set and the natural numbers.

$A$ is enumerable $\iff \exists$ an injection $\sigma: A\to \mathbb{N} \iff A$ is countable

I am not sure if this is true if $A$ is not infinite. Can finite sets me enumerable?

Thank you

Solution 1:

Maybe check the wikipedia page for "Countable Sets". Usually "enumerable" refers to countable (in the not infinite sense).

Note that if there is an injection (that is not a surjection) from a set $A$ to $\mathbb{N}$, this is the same as saying there is a bijection between $A$ and a subset of $\mathbb{N}$, in this case $A$ has at most the cardilanity of the naturals. If this map is also a surjection, then $A$ has the same cardinality as the natural numbers.

The problem in your definition is that it doesn't specify if the map $\sigma$ is strictly an injection or if it can be bijective.