Answer
$(P(1) \lor P(2) \lor P(3))$ $\land$ $\neg$ $ (P(1) \land P(2))$ $\land$ $\neg$ $ (P(2) \land P(3))$ $\land$ $\neg$ $ (P(1) \land P(3))$
Work Step by Step
$\exists ! x P(x)$ means that there exists exactly one x such that P(x) is true.
Domain: 1,2,3
So, we have the following cases:
1) P(1) is True
$\quad$ P(2) is false.
$\quad$ P(3) is false.
2) P(1) is false
$\quad$ P(2) is True.
$\quad$ P(3) is false.
3) P(1) is false
$\quad$ P(2) is false.
$\quad$ P(3) is True.
This means that no two of P(1), P(2), P(3) can be simultaneously true.
So, the answer is
$(P(1) \lor P(2) \lor P(3))$ $\land$ $\neg$ $ (P(1) \land P(2))$ $\land$ $\neg$ $ (P(2) \land P(3))$ $\land$ $\neg$ $ (P(1) \land P(3))$
