What is the equivalent class of $\{a_1,...,a_n\}$

Question

Let $F$ collection of all finite sets, Define $\approx$ on $F$ by $$X\approx Y \iff \exists f:X\to Y\quad\text{s.t. }f\text{ is a bijection}$$

What is the equivalent class of $\{a_1,...,a_n\}$ for distinct $a_1,...,a_n$?

First of all $\approx$ is an equivalence relation on $F$, since I don't know the answer, I've tried to plug some sets and see how it goes $$[A]=\{X\mid \exists f:A\to X\}$$ Hence $$[\{1,2\}]=\{X\mid \exists f:\{1,2\}\to X\}$$

Then $\{1,2\}\in [\{1,2\}]$ and $\{2,4\},\{0.5,1\},...$ so basicaly there is an infinte number of sets just by using the function $f(x)=nx$ where $n\in \mathbb R$, so is that the general answer ?$$[\{a_1,...,a_n\}]=\{k\in \mathbb R\mid \{ka_1,...,ka_n\}\}$$

Answer 1

$a_1, \ldots, a_n$ do not need to be numbers; they just need to be distinct objects. So your analysis leading to $[\{a_1, \ldots, a_n\}] = \{k \in \Bbb R : \{ka_1, \ldots, ka_n\}\}$ cannot make sense in general.

Instead, think about what $X \approx \{a_1, \ldots, a_n\}$ means. It means there is a bijection between $X$ and $\{a_1, \ldots, a_n\}$ i.e. $X$ has the same cardinality as $\{a_1, \ldots, a_n\}$ and obviously the cardinality of $\{a_1, \ldots, a_n\}$ is $n$.

So $[\{a_1, \ldots, a_n\}]$ is the just the collection of all finite sets of size $n$.