I'm trying to find a general propositional formula with $n$ number of variables where its models (the interpretation that satisfies the formula) are exactly those that satisfy an even number of atoms. Also, there is a condition where the formula contains only one occurrence of each atom. e.g
$A \land B \land C$ is true only using the following interpretation
{$A \mapsto 1, B \mapsto 1, C \mapsto 0$} {$A \mapsto 0, B \mapsto 1, C \mapsto 1$} {$A \mapsto 1, B \mapsto 0, C \mapsto 1$}
because there is an even number of 1's.
The things I tried:
I tried building the formula using the truth table. However, if I did that I'll have more than an occurrence per atom.
Also, I tried a more desperate measure of using all variations of connectives (and, or, implies, and double implication only) but still no luck.
I think there will be two formulae for even or odd number of propositional variables, moreover, after searching the Web some places call this parity check formula
The $\leftrightarrow$ is associative, and thus we can generalize it to talke on any number of terms. When you do so, you find that a generalized $\leftrightarrow$ is true iff an even number of its terms are false.
So, we can take:
$$\neg A \leftrightarrow \neg B \leftrightarrow \neg C$$
With $n$ atomic variables $P_1$ through $P_n$, you get:
$$\neg P_1 \leftrightarrow \neg P_2 \leftrightarrow ... \leftrightarrow \neg P_n$$