Kenneth Rosen Edition 7 Exercise 1.6 Question 34 (Page No. 80)

Question

The Logic Problem, taken from WFF’N PROOF, The Game of Logic, has these two assumptions:1. “Logic is difficult or not many students like logic.”2. “If mathematics is easy, then logic is not difficult.”By translating these assumptions into statements involving propositional variables and logical connectives, deter-mine whether each of the following are valid conclusions of these assumptions:

1. That mathematics is not easy, if many students like logic.
2. That not many students like logic, if mathematics is not easy.
3. That mathematics is not easy or logic is difficult.
4. That logic is not difficult or mathematics is not easy.
5. That if not many students like logic, then either mathematics is not easy or logic is not difficult.

Answer 1

1. That mathematics is not easy, if many students like logic. This is not a valid conclusion based on the given assumptions. The first assumption states that "logic is difficult or not many students like logic", and the second assumption states "If mathematics is easy, then logic is not difficult." There is no direct relationship between whether many students like logic and whether mathematics is easy.

2. That not many students like logic, if mathematics is not easy. This is not a valid conclusion based on the given assumptions. The first assumption states that "logic is difficult or not many students like logic", and the second assumption states "If mathematics is easy, then logic is not difficult." There is no direct relationship between whether mathematics is easy and whether not many students like logic.

3. That mathematics is not easy or logic is difficult. This is a valid conclusion based on the given assumptions. The first assumption states that "logic is difficult or not many students like logic", and the second assumption states "If mathematics is easy, then logic is not difficult." Therefore, if mathematics is easy, logic is not difficult, and if logic is difficult then mathematics is not easy.

4. That logic is not difficult or mathematics is not easy. This is a valid conclusion based on the given assumptions. The first assumption states that "logic is difficult or not many students like logic", and the second assumption states "If mathematics is easy, then logic is not difficult." Therefore, if mathematics is easy, logic is not difficult, and if logic is difficult then mathematics is not easy.

5. That if not many students like logic, then either mathematics is not easy or logic is not difficult. This is a valid conclusion based on the given assumptions. The first assumption states that "logic is difficult or not many students like logic", and the second assumption states "If mathematics is easy, then logic is not difficult." If not many students like logic, it can be assumed that logic is difficult, so mathematics is not easy.