Sufficient condition to be a countable set

elementary-set-theory

Let $A$ a non empty set. Which of the following conditions is sufficient because A is a countable set:

  1. $A \subset \mathbb N$

  2. $\mathbb N \subset A$

  3. exists a surjective function $f: \mathbb N \rightarrow A$ and $A$ is infinite

  4. exists a succession $a_n$ so that $\{a_n:n \in \mathbb N\}=A$

My attemp:

  1. is false because if the set $A$ is finite I can't find a bijective function fraom $\mathbb N$ to $A$

  2. is false because if I take $A=\mathbb R$ is not countable

but for the other cases?

Best Answer

  1. It is true. For each $a\in A$, let $n_a=\min f^{-1}(\{a\})$. Then $\Bbb N_A=\{n_a\mid a\in A\}$ is an infinite set of natural numbers and the map from $A$ to $\Bbb N_A$ defined by $f(a)=n_a$ is a bijection. Furthermore any infinite subset of $\Bbb N$ is countable.
  2. It is false. If $A=\{1\}$, take $a_n=1$ form each natural $N$.
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