- #1

rainwyz0706

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Let L = {P } be a ﬁrst-order language with a binary relation symbol

P as only non-logical symbol. By exhibiting three suitable L-structures prove

(informally) that no two of the following sentences logically implies the other

(i) ∀x∀y∀z(P (x, y) → (P (y, z) → P (x, z))),

(ii) ∀x∀y(P (x, y) → (P (y, x) → x = y)),

(iii) (∀x∃yP (x, y) → ∃y∀xP (x, y)).

I really don't have a clue how to handle this problem. Could anyone please give me some hints of the L-structures? Any help is appreciated!