State division and remainder algorithm. Suppose that the domain of the propositional function P(x) consists of the integer 0, 1, 2, 3 and 4. Write out each of the following propositions using disjunctions, conjunctions and negations.

1. ∃x P(x)
2. ∀x P(x)
3. ∃x ¬P(x)
4. ∀x ¬P(x)
5. ¬∃x P(x)
6. ¬∀x P(x)