$1.$

A$)$ What is Propositional logic? Explain the resolution in Propositional logic.

B$)$ Given a set of propositional clauses, prove or disprove the following statements using resolution: $"$If it is raining or it is cloudy, then the picnic will be canceled unless the park has a covered area."

C$)$ Using resolution, determine the validity of the following argument: $"$If John is not at home, then the lights are off. The lights are off. Therefore, John is not at home."

D$)$ Consider a knowledge base consisting of propositional clauses representing a Sudoku puzzle. Use resolution to find a solution to the Sudoku puzzle if one exists.

E$)$ You have a set of propositional clauses representing the behavior of a complex system. Using resolution, determine whether the system will enter a critical state if certain conditions are met.