Our official system of propositional logic, as we developed it in class: . The symbols of the language are: 1. The atomic formulas: A, A2, A3, ... 2. The connectives: and . 3. The grouping symbols: (and). The formulas of the language are defined as follows: 1. Every atomic formula is a formula. 2. If a is a formula, so is (-a). 3. If a and 3 are formulas, so is (a B). 4. Nothing else is a formula. The logical axioms are defined as follows. Suppose a, 6, and y are any formulas of the language. Then the following are logical axioms: A1. (a ( a)) A2. ((a ()) ((a B) (a y))) A3. (((B) (a)) (((B) a) B)) The sole rule of prodecure is Modus Ponens (MP): Given formulas of the language a and (a 3), we may infer 3. If I is a set, possibly empty, of formulas of the language and a is a formula of the language, then a deduction or proof of a from the set of hypotheses I, written as I a (or as a if I = 0), is a finite sequence of formulas of the language, say 91, 92, ..., Pn, with Pn being a, such that each kis 1. a hypothesis, i.e. k EI, or 2. a logical axiom, or 3. follows from preceding formulas in the sequence by Modus Ponens, i.e. there are i, j k such that ; is (ik). 2. Construct deductions in (our version of) propositional logic from the given hypotheses to the given conclusions in each of the following cases: a. Hypotheses: (a ) and (B 7); Conclusion: (a ). [3] b. Hypotheses: none; Conclusion: (((y)) y). [3]