4. Prove that each argument is valid by defining each proposition with a variable to obtain the form of the argument, or defining defining predicates and expressing the hypotheses and the conclusion using the predicates. Then use the rules of inference to prove that the form is valid.

(a) If I drive on the freeway, I will see the fire. I will drive on the freeway or take surface streets (or both). I am not going to take surface streets. I will see the fire.

(b) If it was not foggy or it didnt rain (or both), then the race was held and there was a trophy ceremony. The trophy ceremony was not held. It rained.

(c) Assume the domain is the set of students at an elementary school. Every student who has a permission slip can go on the field trip. Every student has a permission slip. Every student can go on the field trip.

(d) Assume the domain is the people in this world. Everyone who goes to France visits the Louvre. Bob has been to France. Bob has visited Louvre.