The propositions p, q, r, and s are defined by p: Alice is a mathematician. q: Alice loves to solve puzzles. r: Alice has won a math competition. s: Alice enjoys teaching math. (a) Using the definitions above: (i) Express the following in English. (b) S-A ((bvd)) (ii) Express the following in symbolic form. 'Alice has won a math competition if and only if she is a mathematician who does not enjoy teaching math or she loves to solve puzzles.' Construct the truth table for (pq) Ap. Hence determine whether this proposition is a tautology, a contradiction or a contingency. (c) Prove, using the laws of logical equivalence: (p^-q) q = T