Consider a proof system P in the style of box-proof which only have one logical connective A. The system only has one rule given by:

Find the errors and identify them. Put ‘x’ next to the line with errors and the number of the errors using the List of Errors in Predicate Logic Proofs document. (x) (y) Fxy > (Ez) Hxz                          Premise (Ev) Hxv                                                      Premise Hxx                                                               Existential Instantiation (2) (y) Fxy > (Ez) Hxz                                       Universal Instantiation (1) Fxy > (Ez) Hxz                                           Universal Instantiation (4) Fxy > Hxv                                                     Existential Instantiation (5) Fxy                                                               Modus Ponens (3,6) (Ex) Fxx                                                       Existential Generalization (7) (Ex) Fxx v ~ (Ex) Fxx                                  Add (8) (Ex) Fxx                                                       Disjunctive Syllogism (8,9)

Assume the propositions p, q, r, and s have the following truth values:p : False (F). q : True (T). r : False (F). s : True (T). Evaluate the truth value for each of the following compound propositions. Show your work for yourevaluation.a) (p ∨ q) ∧ ¬ rb) p ∨ q ∧ ¬ rc) ¬(r ∨ s) → p ∨ qd) r ⊕ ¬s ↔ q → pe) (p ∧ ¬q) ⊕ (r → (¬s ∨ q))

For each of the attached probabilistic expressions, please answer "yes" or "no" to indicate if it is equal to P(A,B,C), given Boolean random variables A, B and C , and no independence or conditional independence are assumed between any of them.