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); 1(P): {, } Invalid. D:{1}; I(P): { [C] 10.3.2 3x(x +a (Fx v Gx)) Fr 3x((Fx v Gx) +x+a) [A] [B] Valid. Invalid. D:{1}; I(F): {1}; I(G): {1}. Invalid. D:{1, 2}; I(F): {1}; I(G): {2}. 10.3.3 VxVy((Fx Fy) x=y) FT VxVyz(((Fx A Fy) A Fz) ^-((x+y Ay #z) AX#)) [A] [B] [C] Valid. Invalid. D:{1, 2, 3, ...}; I(F): {1, 2, 3, ...} Invalid. D:{1}; I(F): {1} 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); 1(P): {, } Invalid. D:{1}; I(P): { [C] 10.3.2 3x(x +a (Fx v Gx)) Fr 3x((Fx v Gx) +x+a) [A] [B] Valid. Invalid. D:{1}; I(F): {1}; I(G): {1}. Invalid. D:{1, 2}; I(F): {1}; I(G): {2}. 10.3.3 VxVy((Fx Fy) x=y) FT VxVyz(((Fx A Fy) A Fz) ^-((x+y Ay #z) AX#)) [A] [B] [C] Valid. Invalid. D:{1, 2, 3, ...}; I(F): {1, 2, 3, ...} Invalid. D:{1}; I(F): {1}