I. Apply indirect proof to the following arguments. Use the implication rules and the replacement rules. You can also use conditional proof, if needed. [1] 1. P- ~(PvQ) | ~ P Answer: 1. P= ~ (Pv Q) | ~ P 2. P Assumption (IP) 3. ~ (PV Q) 1, 2, MP 4. ~ P . ~ Q 3, DM 5. ~ P 4, Simp 6. P . ~ P 2, 5, Conj 7. ~ P 2-6, IP[2 1. P 1 QV- Q [3] 1. P- (Q . S) 2. ~ S / ~ P [4] 1. PDQ 2. ROP 3. ~ Q | ~ R [5] 1. ~ QVP 2. ~ (PVS) 1 ~ Q 6] 1. (Q - Q)=s / S [7] 1. Pv (~ P- Q) 2. ~ Q / P [8] 1. PDQ 2. PV (Q. S) 1Q [9] 1. [P= (Q . R) ] . (SL) 2. S /L [10] 1. (PV ~ P) - - Q 2. R- Q / ~ R [11] 1. (RV S) = (~ P. ~ Q) 2. P | ~ R [12] 1. RVS 2. Q- ~ R 3. P- Q 4. ~ S | ~ P [13] 1. ~ P- ~ (Qv ~ P) / P [14] 1. PV (Q . P) 2. POR / RII. First, translate the following arguments into symbolic form. Second, use the implication rules, the replacement rules, and indirect proof to derive the conclu sion of each. Letters for the simple statements are provided in parentheses and can be used in the order given. 1. My car is not fuel-efficient and it is not reliable. Consequently, my car is fuel- efficient if and only if it is reliable. (F, R) Answer: 1. ~ F . ~ R IF = R 2. ~ (F = R) Assumption (IP) 3. ~ [ (F . R) V (~ F . ~ R) ] 2, Equiv 4. ~ ( F . R) . ~ ( ~ F . ~ R) 3, DM 5. ~ (~ F . ~ R) 4, Simp 6 . ( ~ F . ~ R ) . ~ ( - F . ~ R ) 1, 5, Conj 7. ~ ~ (F = R) 2-6, IP 8. F = R 7, DN 2. If she finished her term paper on time, then she does not have to work on it over spring break. Either she does have to work on it over spring break or she did not finish her term paper on time and she gets a lower grade. Therefore, she did not finish her term paper on time. (F, S, L) 3. If the murder happened in the hotel room, then there are bloodstains somewhere in the room. It follows that it is not the case that the murder happened in the hotel room and there are not bloodstains somewhere in the room. (M, B) 4. If criminals are not put on trial, then they are likely to commit worse crimes. If criminals are put on trial and they are acquitted, then they are likely to commit worse crimes. Since criminals are acquitted, we can conclude that criminals are likely to commit worse crimes. (T, W, A) 5. It is not the case that Sam did not get the job offer and he is still working at the factory. If Sam did not get the job offer, then he is still working at the factory. We can infer that Sam did get the job offer. (J, F)