Please solve properly with correct reasoning

2. Show that the following arguments are valid using rules of inference. is used for horizontal record of arguments: statements to the left of are premises and statement to the right of is a conclusion. ( 20 pts) (a) pq,q(rs),r(wu),pwu (b) (pq)(rs),rw,wp (c) pq,qr,rp (d) ur,(rs)(pw),q(us),wqp 3. Find an error in the following proof and explain why it is an error. (20 pts) (a) Proving P(x)xP(x) (1) P(x)-premise (2) xP(x) - universal generalization (b) Proving xP(x)yP(y) (1) xP(x) - premise (2) P(y) - existential instantiation ( y in a new free variable name) (3) yP(y) - universal generalization (c) Proving x(P(x)Q(x)),x(P(x)R(x))x(R(x)Q(x)) (1) x(P(x)Q(x))-premise (2) P(c)Q(c) - universal instantiation (c is a constant) (3) x(P(x)R(x)) - premise (4) P(c)R(c) - existential instantiation (5) P(c) - conjunction elimination (6) Q(c) - modus ponens for (2) and