1. For part (a): - Given: Linda owns a red convertible. - Given: Everyone who owns a red convertible has gotten at least one speeding ticket. - By universal instantiation, "Linda" fits the "everyone" category. - By modus ponens on "Linda owns a red convertible" and the universal statement, conclude "Linda has gotten at least one speeding ticket." - By existential generalization, conclude "Someone in this class has gotten a speeding ticket." 2. For part (b): - Given: Five roommates Melissa, Aaron, Ralph, Veneesha, and Keeshawn each have taken discrete mathematics. - Given: Every student who has taken discrete mathematics can take algorithms. - By universal instantiation applied to each roommate and modus ponens, conclude each can take algorithms. - By conjunction, conclude all five can take a course in algorithms. 3. For part (c): - Given: All movies produced by John Sayles are wonderful (universal statement). - Given: John Sayles produced a movie about coal miners. - By universal instantiation, this specific movie is wonderful. - By existential generalization, there is a wonderful movie about coal miners. 4. For part (d): - Given: Someone in this class has been to France (existential statement). - Given: Everyone who goes to France visits the Louvre (universal statement). - By existential instantiation, take that someone who has been to France. - By universal instantiation and modus ponens, conclude that person visited the Louvre. - By existential generalization, conclude that someone in this class has visited the Louvre. Each step uses standard rules of inference: universal instantiation, existential instantiation, modus ponens, conjunction, and existential generalization.