1. Prove using inference rules and laws of logic using at least two different ways. Mark each step with the law or rule you used. If Bob drank coffee last night then he did not sleep well If Bob didn't sleep well, then he is tired today . Bob is not tired today ... Bob didn't drink coffee last night 2. Prove that Superman doesn't exist. From Kalishand and Montague (1) If Superman were able and willing to prevent evil, he would do so (2) If Superman were unable to prevent evil, he would be impotent (3) If he were unwilling to prevent evil, he would be malevolent (4) Superman does not prevent evil (5) If Superman exists, he is neither impotent nor malevolent Therefore, Superman does not exist. Define propositions: M - Superman is malicious 1 - Superman is impotent A- able to prevent evil W-willing to prevent evil P - Superman prevents evil E- Superman exists 3. Prove that no integer can be simultaneously even and odd (proof by contradiction.)