Here is a model: Domain: {1, 2, 3, 4} B : {3, 4}, C : {1, 4}, F : ∅ Make a proposition that is false in this model by filling in each of the blank spaces of （blank)z((B（blank)（blank) z) → （blank)（blank)y¬F y) with one symbol of MPL. Explain why the proposition you made is false in the given model.

Give an example of a model that has exactly two objects in its domain and on which all of the following four propositions are true.∃y∃x(x ≠ y ∧ Syxy) → ∀z(P z → Qz) ∀x∀y∀z(Sxyz ↔ (x = z ∧ y = a))∀x(Qx → x = b) a ≠ b ∧ (Qb ↔ ∃xSxbx)

Consider the following saturated open path and model.∃x¬(x = a → Lxx) X c ∀y(¬Lay ∧ Ryb) \abc¬Laa ∧ Rab X¬Lab ∧ Rbb X¬LaaRab¬LabRbb¬(c = a → Lcc) X c = a ¬Lcc¬Lac ∧ Rcb X¬LacRcb¬Lca¬Lcb↑Domain: {1, 2, 3}Referents: a : 1 b : 2 c : 1Extension of R : {h1, 2i ,h 2, 2i ,h 1, 2i}Extension of L : ∅Are all of the propositions on the open path true on the given model? Explain your answer.

Consider the following two propositions.∀x∀y∀z((x 6 = y ∧ y 6 = z ∧ x 6 = z) → (¬P x ∨ ¬P y ∨ ¬P z))∃z(a = a → ∀yP y)(i) What is the maximum number of objects a model can have in its domain if both of the propositions are true on that model? Explain your answer.(ii) Give an example of a model that has the maximum number of objects in its domain and on which both of the propositions are true.

Here is a model:Domain: {1, 2, 3, 4, ...}Referents: a : 64 b : 2 c : 111Extension of S : {1, 4, 9, 16, 25, ...}Extension of P : ∅Extension of D : {h1, 2i ,h 2, 4i ,h 3, 6i ,h 4, 8i ,h 5, 10i ...}(So for every number 1, 2, 3, ... in the domain, the extension of D contains an ordered pair with that number in first position and the number twice its size in second position.)State whether the following propositions are true or false on the model, and briefly explain your answers.(i) ∀x((Sx ∧ ∃wDwx) → x = a)(ii) ∃z(z 6 = b ∧ P z) ↔ ∃x∀y(Dyx ∨ Dxc)

A builder will build five houses in New Housing Scheme on a street that currently has no houses. The builder will select from seven different models of houses: L,M,N,O,P,Q and R. The Development Authority has placed the following restrictions on the builder: No model can be selected for more than one house. Either model O must be selected or model R must be selected, but both cannot be selected. If model Q is selected, then model N cannot be selected. If model M is selected, then model O cannot be selected. Which of the following is an acceptable combination of models that can be selected for the street? (A) L, M, N, P, Q (B) L, M, P, Q, R (C) L, N, P,Q, R (D) M, N, O, P, Q (E) N, O, P, Q, R