(7) The Logic Problem has two assumptions: (i) "Logic is difficult or not many students like logic." (ii) "If mathematics is easy, then logic is not difficult." By translating these assumptions into statements involving propositional variables and logical connectives, determine whether each of the following are valid conclusions of these assumptions: (a) That mathematics is not easy, if many students like logic. (b) That not many students like logic, if mathematics is not easy. (c) That mathematics is not easy or logic is difficult. (d) That logic is not difficult or mathematics is not easy. (e) That if not many students like logic, then either mathematics is not easy or logic is not difficult.

SOLUTION

Let's assume:

p: Logic is difficult

q: Many students like logic

r: Mathematics is easy

The two given assumptions can be translated in mathematical symbols using the above interpretations

Step Reason

1. p V ¬q Premise
2. r → ¬p Premise
3. ¬r ∨ ¬p Logical equivalence (I) from (2)
4. ¬(r ∧ p) De Morgan's law from (3)
5. ¬p ∨ ¬r Commutative law from (3)
6. p → ¬r Logical equivalence (I) from (5)
7. ¬q ∨ ¬r Resolution from (1) and (3)
8. q → ¬r Logical equivalence (I) from (7)
9. ¬r ∨ ¬q Commutative law from (7)
10. r → ¬q Logical equivalence (I) from (9)
11. (p ∨ ¬q) ∧ (¬r ∨ ¬q) Conjunction from (1) and (9)
12. ¬q ∨ (p ∧ ¬r) De Morgan's law from (11)
13. q → (p ∧ ¬r) Logical equivalence (I) from (12)

(a) "Mathematics is not easy, if many students like logic" can be represented mathematically as

q → ¬r . Since the proposition q → ¬r is mentioned in step (8), the conclusion is valid.

(b)"Not many students like logic, if mathematics is easy" can be represented mathematically as

r → ¬q. The proposition r → ¬q is mentioned in step (10) and is not logically equivalent to ¬r → ¬q therefore making this conclusion invalid.

(C) "Mathematics is not easy or logic is difficult" can be represented mathematically as ¬r ∨ p, In step (3) proposition ¬r ∨ ¬p is mentioned and ¬r ∨ p is not logically equivalent with ¬r ∨ ¬p, thus the conclusion is invalid.

(d) "Logic is not difficult or mathematics is not easy" can be represented mathematically as

¬p ∨ ¬r. The proposition ¬p ∨ ¬r is mentioned in step (5), thus the conclusion is valid.

(e) "If not many students like logic, then either mathematics is not easy or logic is not difficult" can be represented mathematically as ¬q → (¬p ∧ ¬r). We can see that proposition q → (p ∧ ¬r) is mentioned in step (13) and q → (p ∧ ¬r) is not logically equivalent to ¬q → (¬p ∧ ¬r) therefore this conclusion is invalid.