za účelem simulace FULL JOIN v mysql potřebujete UNION výsledek Right Join a Left Join z obou stolů
SELECT Persons.Name, Persons.P_Id, Invoice.Id
FROM Persons
LEFT JOIN Invoice
ON Persons.P_Id=Invoice.P_Id
UNION
SELECT Persons.Name, Persons.P_Id, Invoice.Id
FROM Invoice
LEFT JOIN Persons
ON Persons.P_Id=Invoice.P_Id