So I'm studying for a final- and one of the study questions is "Express (as simply as you can) each of the following sentences without the use of universal quantification:"
And I'm stuck- the textbook mentions nothing here. At first I thought that maybe it was a tricky set of negations, but I think that would just leave me with (∃x)(∀y)(∃z)[¬P(x,y,z)].
The solution to the problem is apparently ¬(∃x)¬(∃y)¬(∃z) ¬P(x,y,z)
So far the closest I can think is to go from
(∀x)(∃y)(∀z)[P(x,y,z)] (Start) ¬( (∀x)(∃y)(∀z)[P(x,y,z)] ) (Negate the whole thing) (∃x)(∀y)(∃z)[¬P(x,y,z)] (Thus swap all quantifiers, negate the inside) ¬( (∃x)(∀y)(∃z)[¬P(x,y,z)] ) (Negate Everything again) ¬(∃x)(∃y)(¬∃z)[P(x,y,z)] (Instead of swapping existential quantifiers, negate them. But we still have no negation on y, and we had to negate the negation on the inside?)