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)