A set of formulas that classifies two-element structures

Question

Give a set of formulas $\Gamma$ such that for any structure $\mathcal{A}=\langle A;-;-\rangle$ it holds that $\mathcal{A} \models \Gamma$ if $A$ has exactly two elements.

Answer 1

The formula you mention the comments works, but should be parenthesised as follows: $$\exists x \exists y \forall z ((z=x \vee z=y) \wedge x \ne y)$$ It says there are two elements which are not equal such that any element is one or the other... which is precisely the statement that there are exactly two elements.

In general it's not true that $(A \vee B) \wedge C \equiv A \vee (B \wedge C)$, and without including parentheses it's unclear which of these $A \vee B \wedge C$ is taken to mean.