10.3 Identity: Truth Trees. Each question is worth 3 marks. Construct a truth tree to determine whether the following sequents are valid or invalid. For each invalid sequent, construct a counter-model. 10.3.1 Vx(Pxa Pxb), (Pca a Vy(Pyb y= c)) FT (3xPxa Vy(Pya+y=c)) [A] [B] Valid. Invalid. D:{1, 2, 3}; I(P): {, } Invalid. D:{1}; I(P): {} [C] 10.3.2 3x(x + a + (Fx v Gx)) FT 3x((Fx v Gr) x #a) [A] Valid. [B] Invalid. D:{1}; I(F): {1}; I(G): {1}. Invalid. D:{1, 2}; I(F): {1};I(G): {2}. [C] 10.3.3 VxVy((Fx Fy) ^ x = y) Et VxVyVz(((Fx ^ Fy) ^ Fz) ^-((xy^y #z) ^ x #z)) Valid. [A] [B] [C] Invalid. D:{1, 2, 3, ... }; I(F): {1, 2, 3, ... } Invalid. D:{1}; I(F): {1} 10.3 Identity: Truth Trees. Each question is worth 3 marks. Construct a truth tree to determine whether the following sequents are valid or invalid. For each invalid sequent, construct a counter-model. 10.3.1 Vx(Pxa Pxb), (Pca a Vy(Pyb y= c)) FT (3xPxa Vy(Pya+y=c)) [A] [B] Valid. Invalid. D:{1, 2, 3}; I(P): {, } Invalid. D:{1}; I(P): {} [C] 10.3.2 3x(x + a + (Fx v Gx)) FT 3x((Fx v Gr) x #a) [A] Valid. [B] Invalid. D:{1}; I(F): {1}; I(G): {1}. Invalid. D:{1, 2}; I(F): {1};I(G): {2}. [C] 10.3.3 VxVy((Fx Fy) ^ x = y) Et VxVyVz(((Fx ^ Fy) ^ Fz) ^-((xy^y #z) ^ x #z)) Valid. [A] [B] [C] Invalid. D:{1, 2, 3, ... }; I(F): {1, 2, 3, ... } Invalid. D:{1}; I(F): {1}