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😃: Express the negation of the statement ∀x ∃y (P(x,y) ∧ ∃z R(x,y,z)) so that
all negation symbols immediately precede predicates. ! solve this
🤖: Sure! To express the negation of the statement ∀x ∃y (P(x,y) ∧ ∃z R(x,y,z)), we can use the following formula:

◇∀x ∃y (¬P(x,y) ∨ ¬∃z R(x,y,z))

Here's how it works:

• The symbol "◇" indicates that the negation of the enclosed formula is being expressed.
• ∀x ∃y (P(x,y) ∧ ∃z R(x,y,z)) means "for all x, there exists y such that P(x,y) and z exists where R(x,y,z) holds".
• ¬P(x,y) means "it is not the case that P(x,y)", which corresponds to the first part of the original statement.
• ¬∃z R(x,y,z) means "there does not exist a z such that R(x,y,z)", which corresponds to the second part of the original statement.

So the formula ◇∀x ∃y (¬P(x,y) ∨ ¬ EXISTS z R(x,y,z)) expresses the negation of the original statement.