Find the set of all equivalence classes of the relation $\phi \sim \psi \iff \models(\phi \iff \psi)$

Question

Find the set of all equivalence classes of the relation $\phi \sim \psi \iff \models(\phi \iff \psi)$ All sentences consist of only one variable.

Since these sentences have to be equivalent and each of them consists of only one variable, then there are $2 \cdot 2$ ways to assign the logical values of the sentences to the values of the variables. And so I can think of four equivalence classes. Let $\pi(x)$ be the logical value of a sentence whose variable has the value of $x$. Then to describe my equivalence classes I'd like to use this ordered pair: $(\pi(0), \pi(1))$. And so, the equivalence classes:
$$\{[0,0], [0,1], [1,0], [1,1] \}$$ Is this solution correct?

Answer 1

Yes.

Put a different (maybe a little more intuitive) way: every sentence built up form a single variable $P$ (and any number of logical operators) will be logically equivalent to one of four sentences: $P$, $\neg P$, $\top$, or $\bot$