What is the cardinality of the equivalence class

Question

Consider this relation:

$$R = \left\{ {\left\langle {f,g} \right\rangle \in {{\left\{ {0,1} \right\}}^N} \times {{\left\{ {0,1} \right\}}^N}|\exists k \in N\left| {\left\{ {i \in N|f(i) \ne g(i)} \right\}} \right| = k} \right\}$$

It's actually saying, $\left\langle {f,g} \right\rangle \in R$ iff $f$, $g$ don't agree with each other countable times ($k \in \mathbb{N}$).

What is the cardinality of the following equivalence class:
$$\left| {{{\left[ {\lambda n \in N.o} \right]}_R}} \right|$$

Obliviously, it has to be at least $\aleph_0$, We can demonstrate it by the series $1000...,11000...,111000...$

But how to show it's exactly $\aleph_0$?

Answer 1

HINT: The set of finite subsets of $\Bbb N$ is countable.