l. [8 marks] Consider the following compound proposition: ((q >p) A an) rm/q) (a) Work out the truth table for this proposition and explain why your truth table shows that it is an equivalence. (b) Use the fundamental logical equivalenees (Tables 7.67.7 in the Unit Notes, or the 'logic tables' pdf on the LMS) to prove that it is an equivalence. [Hint: You might start by using the contrapositive equivalence] 2. [7 marks] Express the following argument in symbolic form and test its logical validity by hand. If the argument is invalid, give a counterexample; otherwise, prove its validity using the rules of inference. If there is an increase in supply of housing and no increase in demand for it, then the average cost of housing will decrease. If we want to increase the supplyr of houses, then the government must make more land available. There will be no increase in demand, and the government has made more land available, therefore the average cost of housing will decrease. 3. [5 marks] Use mathematical induction to prove that if (tn) is a sequence defined recursively by t1 = 1; t.\" = i.,_1+3n23n+1, n 2 2. then tn = n3 for all integers n 2 1. 4. [5 marks] Use the axioms and fundamental identities of Boolean algebra to simplify the following Boolean expression (showing all the steps): I(y($ + 2W + {93' + y)(ry')