$1.$ Logic and propositional calculus. $($a$)(5$ points$)$ Basic definitions and properties $($propositions and compound propositions; logical operators on propositions; converse proposition; contrapositive proposition; inverse proposition; equivalent propositions; biconditionals; truth tables; precedence of logical operators; consistent system; tautology; contradiction; satisfiable proposition; predicates; quantifiers$).$ Give an exhaustive definition of each notion. $($b$)(5$ points$)$ Rclated theorems $($applications of propositional logic; De Morgan's laws; main logical equivalences; logical equivalences involving conditional and biconditional statements; negating quantified expressions$).$ Demonstrate your understanding of these theorems by examples.