1. Here are some attempts at proofs inspired by answers that students have given in previous semesters. For each proof, I would like you to explain which steps (if any) are logically incorrect and why. Note that I am not asking you to to explain how they "should" have written the proof. I've included some proofs where every step is completely correct, even though the proofs include weird strategies, false starts, or other oddities. These proofs may not be good proofs, and you might know of a better strategy, but that doesn't make the proofs wrong. As long as every step correctly applies a valid inference rule to the available formulas, then you should state that the proof is correct. I've included line numbers just to make it easier to talk about the different steps. To give you an idea of what I'm looking for, here's an example of a proof and the kind of answer I want from you: Proof. 1. Assume (FG)G and F. 2. Assume FG. 3. Since F and FG are true, so is G. (Appl.) 4. Because we have F and G, we can conclude FG. (^-Intro.) Answer: Line 4 is wrong because you can't use formulas from a subproof after the subproof has ended. Finally, note that there are no mistakes in format, phrasing, citations, or other aspects of presentation, so don't worry about that sort of thing. Just pay attention to what formulas are being used, what rules are being used, and what formula is being concluded. (a) Proof. 1. Assume (FG)G and F. 2. Since (FG)G is true, so is G. (V-Elim.) 3. From G, we can derive G. (Dbl. Neg.) 4. Because we have F and G, we can conclude FG. ( -Intro.) (b) Proof. 1. Assume AC and B. 2. Assume A. 3. From A and B, we get AB. (^-Intro.) 4. Because A and AC, we can conclude C (Appl.) 5. Since C, we know C. (Dbl. Neg.) 6. Since we have AB and C, we know (AB)C (Dir. Pf.) (c) Proof. 1. Assume YZ and XY. 2. From YZ, we can derive Y and Z (^-Elim.) 3. Suppose towards a contradiction that X is true. 4. From X and XY, we get Y. (Appl.) 5. But Y contradicts our earlier statement Y. 6. Therefore X. (contrad.) 7. Suppose towards a contradiction that XZ is true. 8. Since XZ, we get X. (^-Elim.) 9. But X and X can't both be true. 10. Therefore we have (XZ). (contrad.) (d) Proof. 1. Assume YZ and XY. 2. Suppose towards a contradiction that X is true. 3. Because X and XY hold, so does Y. (Appl.) 4. Since YZ, we have Y and Z (^-Elim.) 5. From X and Z, we can conclude XZ (^-Intro.) 6. It's not possible to have both Y and Y. 7. From XZ, we proved a contradiction, and therefore (contrad.) (XZ)