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Say whether each of the following sentences is valid, contingent, or unsatisfiable.
| a. |
∀x.(p(x)=>∃y.(p(y) & r(x,y))) |
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| b. |
∀x.∃y.(p(x,y) => ∀z.(p(y,z) ⇒ p(x,z))) |
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| c. |
∀x.(p(x,x) ⇒ ∃y.(p(x,y) ∧ p(y,x))) |
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| d. |
∀x.∀y.(p(x,y) ⇒ ∀z.(p(y,z) ∧ p(z,x))) |
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| e. |
(∀x.(p(x) ⇒ ∃y.q(x,y)) ∧ (∃x.∀y.(¬q(x,y) ∧ p(x))) |
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