(a)∀z∃y∃x¬T(x,y,z)
(b)Obtain the negation by steps:
1)¬(∃x∃yP(x,y))∨¬(∀x∀yQ(x,y))
2)∀x∀y¬P(x,y)∨∃x∃y¬Q(x,y)
(c)By steps:
1)∀x∀y¬(Q(x,y)↔Q(y,x))
2)∀x∀y(Q(x,y)⊕Q(y,x)), where ⊕ is exclusive or
(d)By steps:
1)∃y∀x∀z¬(T(x,y,z)∨Q(x,y))
2)∃y∀x∀z(¬T(x,y,z)∧¬Q(x,y))
Answer:
(a)∀z∃y∃x¬T(x,y,z)
(b)∀x∀y¬P(x,y)∨∃x∃y¬Q(x,y)
(c)∀x∀y(Q(x,y)⊕Q(y,x))
(d)∃y∀x∀z(¬T(x,y,z)∧¬Q(x,y))
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