I. Use the first four implication rules to complete the proofs. Provide the justifi- cation for each step that you derive. [1] 1. ~ (P. Q) 2. ~ ( R . S ) = ( L . ~ Q) 3. ( R . S) - (P . Q) / L . ~ Q Answer: 4 . ~ (R . S ) 1, 3, MT 5. L . ~ Q 2, 4, MP [2] 1. PDQ 2. ROP 3. ~ Q / ~ R [3] 1. P 2. R- Q 3. P- ~ Q / ~ R [4] 1. S= (P. Q) 2. (P . Q) -R 3. ~ R 1 ~S [5] 1. ~ P- (QV R) 2. ( ~ P - ~ S ) - ~L 3. (QV R) - ~ S / ~ L\fII. First, translate the following arguments into symbolic form. Second, use the four implication rules to derive the conclusion of each. Letters for the simple state- ments are provided in parentheses and can be used in the order they are given. 1. Shane is going to the party, or either Rachel or Max is going. Either Rachel is going to the party or Shane is not going to the party. But Rachel is not going to the party. Therefore, Max is going. (S, R, M) Answer: 1. Sv(RvM) 1 RV~S 3. ~R /M 4. ~S 2,3,DS 5. RVM 1,4,DS 6. M 3,5,DS I If I bet red on roulette, then I will win my bet. If I win my bet, then I will stop betting. If I'm feeling lucky, then Ibet red on roulette. I'm feeling lucky. It follows that I will stop betting. (R, VIC S, L) I If Melinda is a comedian, then she is shy. Either Melinda is a comedian, or if she is not shy, then she is famous. Moreover, Melinda is not shy. Consequently, she is famous. (C, S, F) I If we continue to ght, then our supply of troops grows thinner. If our supply of troops grows thinner, then either enlistment slows down or more casualties *5. will occur. But we do continue to ght. Also, enlistment does not slow down. This proves that more casualties will occur. (F, S, E, C) If my son drinks three sodas, then if he eats some chocolate, then he gets hyper. If he is excited, then my son drinks three sodas. Furthermore, my son is excited, or he either drinks three sodas or he eats some chocolate. But it is not the case that if he eats some chocolate, then he gets hyper. We can conclude that he eats some chocolate. (S, C, H, E) If amino acids were found on Mars, then there is life on Mars, then there is life in the universe outside Earth. Either amino acids were found on Mars or we did not look in the best places. If we did not look in the best places, then if amino acids were found on Mars, then there is life on Mars. But it is not the case that amino acids were found on Mars. Thus, there is life in the universe outside Earth. Mano Either I am going to the movie or I am studying for the exam. If I study for the exam, then I will not fail the course. But I either fail the course or I will graduate on time. I am not going to the movie. Hence, Iwill graduate on time. (M, S, F, G) If there is a recession and the housing sector does not recover, then the national debt will continue growing. Also, the government invests in public projects or the national debt will not continue growing. Either there is a recession and the housing sector does not recover, or the unemployment rate will not go down. But the government is not investing in public projects. This implies that the unemployment rate will not go down. (R, H, D, R. U) . If Suzy buys a new car or a new motorcycle, then she has to take a loan. If Suzy saves half her weekly salary for a year, then if she doesn't go on an expensive vacation, then she will not have to take a loan. Either she goes on an expensive vacation or she saves half her weekly salary for a year. But Suzy does not go on an expensive vacation. Therefore, it is not the case that either Suzy buys a new car or a new motorcycle. (C, M, L, S, E) I If your aunt is not a lawyer, then she is an accountant. In addition, if your aunt is an accountant, then if she is tired of her job, then she can teach at our college. Your aunt is either looking for new employment or she cannot teach at our col- le ge. But your aunt is not a lawyer. Also, she is not looking for new employment. Therefore, she is not tired of her job. (L, R I, C, E)