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)

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