Set which is Equivalent to Countable Set is Countable
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Lemma
Let $S$ be countable set.
Let $T$ be a set.
Let $T$ be equivalent to $S$.
Then $T$ is countable.
Proof
By definition of set equivalence:
- $S$ is in one-to-one correspondence with $T$.
We have that $S$ is countable.
By definition of countable set:
- $S$ is in one-to-one correspondence with a subset of the natural numbers.
$T$ is in one-to-one correspondence with $S$.
By Composite of Bijections is Bijection:
- $T$ is in one-to-one correspondence with a subset of the natural numbers.
Hence by definition of countable set: $T$ is countable.
$\blacksquare$