prove in words: If {X, Y, Z} ` A ∧ ¬A, then {Y, Z} ` ¬X ` = derivability

Construct direct proofs to show that the following symbolic arguments are valid. Commas mark the breaks between premises. ~C→(F→C), ~C ∴ ~F

If p is true, q is true, and r is false, determine whether each of the following compound statements aretrue or false”(a) p ∧ q(b) ∼ (q ∨ r)(c) r → (p ∧ q)(d) (p ∧ r) ∨ (p ∧ q → r)

Rewrite each of the following statements so that no negation is outside a quantifier or an expression involving logical operations. a.~ (VxSy ≤ T P(x, y)). b. ~ (Vx € Sy € T P(x, y)). c. ~ (VxSVy ET (P(x, y) V Q(x, y))). d. ~ (Ex € Sy T ~P(x,y) ^xy T, Q(x, y)). Sy e. ~ VxS (y TVzEL P(x, y, z) 32 Ly S, P(x, y, z)).