6:38 6 https://sit.instructure.com/files/46.. CS 135 Spring 2019: Problem Set 2 Problem 1. (20 points) For each of the following use argument is valid, or else find a counterexample. a I5 points) b. 15 points [ 10 points) Hypothesis 1: If Superman were able and willing to prevent evil, he would do so. Hypothesis 2: If Superman were unable to prevent evil, he would be impotent Hypothesis 3: If Superman were unwilling to prevent evil, he would be malevolent Hypothesis 4: Superman does not prevent evil Hypothesis S: If Superman exists, he is neither malevolent nor impotent. Conclusion: Therefore, Superman does not exist C. Problem 2. (15 points) Let Loves(x, y.d) be the predicate "x loves y on day d Thus, for example, 3x d Loves(x"Juliet, d) means that there is someone who loves Juliet every day. a. (1 point) What are the domains of variables x and d in the example above? Express each of the following statements as a quantified predicate. b. [2 points) Every day Juliet is loved by someone. c. (3 points) lago never loves himelf Now, let Future(dy d) denote "day d comes before day da. Also let EQ(x, y) denote x and y are the same person" and Eo(dy,dz) denote "day dis the same as d. Use these in addition to Loves(x.y,d) to express the following statements. d. 3 points) There is a person who, on each day, loves someone other than himsell e. (3 points) Everyone who someday loves a person who loves everyone everyday loves that person ever after f. 3 points, No one loves anyone on days when they are not loved by anyone else. & (Extra Credit S points.) Anyone who loves one person one day, and on a later day lover someone else but not the person he loved the first day is not loved after the later day by the person he loved earlier Dashboard