1. Write the following propositions in the form p → q for appropriate p and q. (a) Willy gets caught whenever he cheats. (b) A necessary condition for Jim to buy a computer is that he obtains $2,000. (c) The website will be accessible only if you have the correct password. (d) You can walk the Appalachian trail unless you can't take 6 months off work. What are the converse and contrapositive of each of the propositions?
2. (a) Construct the truth table for q ↔ (¬p ∨ q). (b) By examining your truth table, find a very simple expression, using p and q, that is logically equivalent to q ↔ (¬p ∨ q). (The ‘very simple expression' should involve just a very small number of connectives such as ∧, ∨, or ¬.) (c) Provide a descriptive explanation for the equivalence. (d) Using the laws of logic show that this restatement is equivalent.