Show that (P → Q) ↔ (¬P ∨ Q) in fitch
To show that (P → Q) ↔ (¬P ∨ Q) is true, we need to prove two directions:
1. (P → Q) → (¬P ∨ Q)
2. (¬P ∨ Q) → (P → Q)
Let's start with direction 1:
1. (P → Q) Assumption
2. ¬(¬P ∨ Q) Assumption (for proof by contradiction)
3. P Assumption
4. Q →E(1,3)
5. ¬P ∨ Q ∨I(4)
6. ¬(¬P ∨ Q) ¬I(5)
7. ¬P Assumption
8. ¬P ∨ Q ∨I(7)
9. ¬(¬P ∨ Q) ¬I(8)
10. ¬(¬P ∨ Q) ∨E(3,5-6,7-9)
11. (¬P ∨ Q) → (¬(¬P ∨ Q)) →I(2,10)
12. (¬P ∨ Q) ∨E(1,11)
13. (P → Q) → (¬P ∨ Q) →I(1,12)
Now, let's move on to direction 2:
1. (¬P ∨ Q) Assumption
2. ¬P Assumption
3. ¬P ∨ Q ∨I(2)
4. (P → Q) →I(3)
5. P Assumption
6. Q ∨E(1,5)
7. (P → Q) →I(6)
8. (P → Q) ∨E(5,7)
9. (¬P ∨ Q) → (P → Q) →I(2,8)
Now, we have proved both directions:
1. (P → Q) → (¬P ∨ Q)
2. (¬P ∨ Q) → (P → Q)
Therefore, (P → Q) ↔ (¬P ∨ Q) is true.