Find a formula involving only the connectives ¬ and → that is equivalent to p ∧ q.

If p=T, q=unknown and r=F, what is p ∨ r ∧ q?

Write the following in disjunctive normal form using logical equivalences( p ∨q) →(p ∧¬r).

CREATE A FSM STATE DIAGRAM USING THE FOLLOWING INFORMATION:The system has four states, which represent the current level of the elevator (inorder from highest level to lowest level): Entrance Floor (ground level), EmployeeParking (first underground level), Underground Parking 2 (one level below em-ployee parking), Underground Parking 3 (two levels below employee parking)•The system has two 1-bit inputs: Up (label: u), Down (label: d)•The system has one 2-bit output: Level (label: L or L1 and L0 for each individualbit, where index 1 indicates the MSB and index 0 the LSB) •If the Up button is pressed, the elevator goes to the level above. If the elevator isalready in the highest level, pressing the Up button will have no effect and will stayin the current level. In case the level above is the employee parking, the elevatorwill skip this level and go directly to the level above it.•If the Down button is pressed, the elevator goes to the level below. If the elevatoris already in…

Determine whether the following proposition is a tautology: (¬p∨¬(r⟶q))⟷(p⟶(¬q∧r)) can i get a non handwriting answer so it would be easy to copy please

A set of formula Γ is consistent iff there is no formula A such that Γ⊢A and Γ⊢¬A. Use the soundness and the completeness of propositional logic/calculus to show that Γ is satisfiable if and only if it is consistent.

Prove Pierce's law: For any propositions P and Q, the proposition ((P -+ Q) -> P) -+ P is a tautology

1.     Show that (p ∧ q) → r and (p → r) ∧ (q → r) is logically equivalent.

State whether each of the following functions is injective, surjective, bijective, or none ofthese categories: (N is the set of natural numbers, and Σ∗is the set of all strings over Σ.)  h : Σ∗ → Σ∗, where Σ = {a, b}, and h(w) = wR.

Simplify the proposition using the equivalence laws (C ↔ B) ∧ (((¬B ∨ A) → ¬B) → ¬C)