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 gallery.