1. Build a combined truth table for the following wffs, where P, Q and R are propositional variables: (a) P ? (Q ? R) (b) (P ? Q) ? R Use your tables to explain briefly why (P ? Q) ? R |= P ? (Q ? R) , but P ? (Q ? R) ?|= (P ? Q) ? R . (8 marks) 2. Use the rules of deduction in the Propositional Calculus (but avoiding derived rules) to find formal proofs for the following sequents:(a) P , (P ? Q) ? ? R ? R ? ? Q(b) Q ? (R ? ? Q) ? ? (R ? Q)(c) (Q ? R) ? (P ? S) ? (P ? Q) ? (R ? S) (12 marks) 3. Use truth values to determine which one of the following wffs is a theorem (in the sense of always being true).(a) ( ( P ? R ) ? ( Q ? R ) ) ? (( P ? Q ) ? R )(b) ( ( P ? R ) ? ( Q ? R ) ) ? (( P ? Q ) ? R )For the one that isnt a theorem, produce all counterexamples. For the one that is a theorem, provide a formal proof also using rules of deduction in the Propositional Calculus (but avoiding derived rules of deduction). (10 marks) 4. A tilde-arrow-wff is a well-formed formula that is built out of propositional variables and the logical connectives ?, ? and ? only. For example W1 ? ((? ((? P) ? Q)) ? ((? R) ? (P ? Q))) is a tilde-arrow-wff. For part (a) below, each bracket and each propositional variable, including its subscript, counts as a symbol. We also include in the count of symbols the final brackets on the outside of the wff that are normally invisible in practice.(a) Prove that the number of symbols of tilde-arrow-wffs can be 1, 4, 5 or any integer greater than or equal to 7. (We saw in lectures that the number of symbols of a tilde-arrow-wff cannot be 2, 3 or 6, and you do not need to prove this.)(b) Use truth values to explain why W1 is a theorem.(c) Find a tilde-arrow-wff W2 that has
the following truth table (including a brief explanation how you found it): P T T T T F F F F Q T T F F T T F F R T F T F T F T F W2 T F F T T F F F(d) Prove that all truth tables, in any number of propositional variables, arise as truth tables of tilde-arrow-wffs. (15 marks) 5. Evaluate each of 1 3 , 7 8 , 8 7 in Z12 and Z13, or explain briefly why the given fraction does not exist. (6 marks) 6. Prove that the only integer solution to the equation x 2 ? 3y 2 = 2z 2 is (x, y, z) = (0, 0, 0). (9 marks)
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