Set 10 Part 1 (one point total). Briefly answer each of the following questions. 1. What are the three rules that allow you to discharge assumptions? 2. What are the restrictions on Universal Generalization (Gen)? 3. What are the restrictions on Existential Proof (Proof)? 4. What is the assumption set for line that uses Identity Introduction (=I)? Part 2 (one point total). For each of the following sequents, provide a proof that demonstrates their validity. 1. x(GxPx), xPxxPx xGxxPx 2. x(GxPx) xGxxPx 3. x(GxPx), x(PxHx) GaHa 4. x(GxPx), x(PxHx) x(GxHx) 5. x(GxPx), x(PxHx) x(HxGx) 6. x((Gx&Px)yLxy), x(Gx&Px) xyLxy 7. x((Gx&Px)yLxy), xy(LxyLyx), x(Gx&Px) xy(Gy&Lxy) 8. x((Gx&Px)yLxy), xy(LxyLyx), Ga&Pb a=byLay 9. x((Gx&Px)yLxy), xy(LxyLyx), Ga&Pb a=by(Lya&Lby) 10. x((Gx&Px)yLxy), xy(LxyLyx), xyz((Lxy&Lyz)Lxz), Ga&Pb a=bxLxx Part 3 (two points total). For each of the following sequents, provide a proof that demonstrates their validity. 1. x((Gx&Px)yLxy), x(Gx&Px) xyLxy 2. x((Gx&Px)yLxy), xy(LxyLyx), x(Gx&Px) xy(Gy&Lxy) 3. x((Gx&Px)yLxy), xy(LxyLyx), Ga&Pb a=byLay 4. x((Gx&Px)yLxy), xy(LxyLyx), Ga&Pb a=by(Lya&Lby) 5. x((Gx&Px)yLxy), xy(LxyLyx), xyz((Lxy&Lyz)Lxz), Ga&Pb a=bxLxx 6. x(Gx&Fx), x(GxHx) xHx 7. x(Gx&Fx), x(GxHx) xHx&yGy 8. x(Gx&Fx), x(GxHx) xy((Hx&Gy)(Fy&Gx)) 9. x(Gx&Fx), x(GxHx), xy(Fx&GyRxy) Faxy((Gx&Fy)&Ryx) 10. x(Gx&Fx), x(GxHx), xy(Fx&GyRxy) xFxxy((Gx&Fy)&Ryx)