(Some translations) Tarski's World provides you with a very useful way to check whether your translation of a given English sentence is correct. If it is correct, then it will always have the same truth value as the English sentence, no matter what world the two are evaluated in. So when you are in doubt about one of your translations, simply build some worlds where the English sentence is true, others where it is false, and check to see that your translation has the right truth values in these worlds. You should use this technique frequently in all of the translation exercises. Start a new sentence ¯le, and use it to enter translations of the following English sentences into ¯rst-order logic. You will only need to use the connectives ^; V; and -. 1. Either a is small or both c and d are large. 2. d and e are both in back of b. 3. d and e are both in back of b and larger than it. 4. Both d and c are cubes, however neither of them is small. 5. Neither e nor a is to the right of c and to the left of b. 6. Either e is not large or it is in back of a. 7. c is neither between a and b, nor in front of either of them. 8. Either both a and e are tetrahedra or both a and f are. 9. Neither d nor c is in front of either c or b. 10. c is either between d and f or smaller than both of them. 11. It is not the case that b is in the same row as c. 12. b is in the same column as e, which is in the same row as d, which in turn is in the same column as a.