1. (6pts) What rule of inference is used in each of these arguments? (a) It is raining today. If it is raining, I will not ride my bike to school. Therefore, I will not ride my bike to school. (b) Liz is a lizard who cannot fly or swim. Therefore, Liz cannot fly or swim. (c) If you have free time and a magical sibling, you can build a snowman. You have free time and a magical sibling. Therefore, you can build a snowman. (d) George is either a spider or has wings. George does not have wings. Therefore, George is a spider. (e) If you cannot fix this pipe on your own, then you need to call a plumber. You do not need to call a plumber. Therefore, you can fix this pipe on your own. (f) If you exercise regularly, then you boost your energy. If you boost your energy, then you feel happier. Therefore, if you exercise regularly, then you feel happier. 2. (5pts) Prove the validity of the following famous arguments. Specify the domain and rules of inference you used. (a) "All men are mortal. Socrates is a man. Therefore, Socrates is mortal." (b) "No man is an island. Manhattan is an island. Therefore, Manhattan is not a man." 3. (24pts) Determine whether or not the following arguments are valid. If they are valid, then show your proof and state the rules of inference used to prove validity. If they are invalid, outline precisely why they are invalid. For arguments with quantifiers assume that the domain is { all humans }. (a) If it is very cold or slippery, the classes will be cancelled. The classes will not be cancelled. Therefore, it is not slippery. (b) I will buy a new car and a new house only if I get a job. I am not going to get a job. Therefore, I will not buy a new car. (c) Mia, a student in COMP 1805 class, plays the violin. Everyone who plays violin owns bow rosin. Therefore, someone in COMP 1805 class owns bow rosin. (d) Not everyone can solve this problem. Someone is very clever. Therefore, there is someone very clever who cannot solve this problem. (e) Everyone is swimming but no one is cold. If it is raining then Michael goes swimming and Margo is cold. Therefore it is not raining. (f) Akiva is an artist and Kaufman owns a gallery. Everyone who owns a gallery is an artist. Therefore, Kaufman is an artist and Akiva owns a gallerv