1. Prove
in five different ways:
(a) Prove it using the method of Section 4.1. That is, either transform the antecedant into the consequent or vice versa. (The consequent has "more structure", so it may be better to start with it.) If you transform antecedant into consequent, then in each step the relationship between the two relevant expressions must be either ≡ or ⇒ (implies). If you go in the opposite direction, the relationship in each step must be either ≡ or ⇐ (is consequence of).
(b) Prove it using the method of assuming the antecedant.
(c) Prove it using the method of proof by case analysis. Use r (i.e., E[r := true]) and ¬r (i.e., E[r := false]) as the two cases. (In other words, prove each of (3.76e)[r:=true] and (3.76e)[r:=false].)
(d) Prove it using the method of proof by contradiction. (To prove A by contradiction, show ¬A ⇒ false.)
(e) Prove it using the method of proof by contrapositive. (To prove A ⇒ B by contrapositive, show ¬B ⇒ ¬A.)
2. Using the methods of Gries & Schneider, prove the validity of the following argument.
Louise is lying. If kangaroos are numerous and Louise is lying, then Harriet is lying. If kangaroos are numerous and the dog got outside last night and Harriet is lying, then I am an idiot. If the dog got outside last night, then kangaroos are numerous. If Louise is lying, then the dog got outside last night. Therefore, I am an idiot.
Don't forget that any boolean expression can be rewritten as any expression to which it is equivalent. Also recall that an implication is equivalent to its contrapositive.
3. Do the previous problem again, but this time using resolution.