1. For part (a): - Problem: "Linda owns a red convertible, and everyone who owns a red convertible has gotten at least one speeding ticket. Prove that someone in this class has gotten a speeding ticket." - Step 1: Linda owns a red convertible. (Given) - Step 2: Everyone who owns a red convertible has gotten at least one speeding ticket. (Given) - Step 3: Therefore, Linda has gotten at least one speeding ticket. (Universal Instantiation applied to step 2) - Step 4: Linda is someone in this class who has gotten a speeding ticket. (Existential Generalization from step 3) 2. For part (b): - Problem: "Each of five roommates has taken a course in discrete mathematics. Everyone who has taken discrete mathematics can take algorithms. Prove all five can take algorithms next year." - Step 1: Melissa, Aaron, Ralph, Veneesha, and Keeshawn have taken discrete mathematics. (Given) - Step 2: Everyone who has taken discrete mathematics can take algorithms. (Given) - Step 3: Melissa can take algorithms (Universal Instantiation applied to step 2 and modus ponens with Melissa) - Step 4: Similarly, Aaron, Ralph, Veneesha, and Keeshawn can take algorithms by the same reasoning. - Step 5: Therefore, all five roommates can take algorithms next year. (Conjunction of step 3 and 4 conclusions) 3. For part (c): - Problem: "All movies by John Sayles are wonderful. He produced a movie about coal miners. Prove there is a wonderful movie about coal miners." - Step 1: All movies by John Sayles are wonderful. (Given) - Step 2: John Sayles produced a movie about coal miners. (Given) - Step 3: This specific movie is a movie by John Sayles (Universal Instantiation from step 2) - Step 4: Therefore, the movie about coal miners is wonderful. (Modus Ponens on steps 1 and 3) - Step 5: Hence, there exists a wonderful movie about coal miners. (Existential Generalization from step 4) 4. For part (d): - Problem: "Someone in the class has been to France. Everyone who goes to France visits the Louvre. Prove someone in this class has visited the Louvre." - Step 1: There exists someone in the class who has been to France. (Given) - Step 2: Everyone who goes to France visits the Louvre. (Given) - Step 3: By Existential Instantiation, pick that someone who has been to France. - Step 4: That person visits the Louvre. (Universal Instantiation and Modus Ponens on step 2) - Step 5: Therefore, someone in this class has visited the Louvre. (Existential Generalization from step 4)