help please anyone?
i'm having a hard time solvig these problems
LOGIC CONDITIONAL PROOF. HOMEWORK HELP PLEASE!?
A.
1. A -> (B -> C)
2. (C ^ D) -> E
3. -F ->(D ^ -E) / :. A -> (B -> F)
B.
1. (A v B) -> -(C ^ D)
2. (-C v -D) -> (E <-> F)
3. (E <-> F) -> (G ^ H) / :. (A v B) -> (H ^ G)
Sure, I can help you solve these logic conditional proof problems.
Let's start with problem A:
1. A -> (B -> C)
2. (C ^ D) -> E
3. -F -> (D ^ -E)
To prove: A -> (B -> F)
To solve this problem, we'll use a conditional proof.
1. Assume A is true.
2. To prove: B -> F
3. Assume B is true (under the assumption of A).
4. To prove: F
5. From 3 and 1, use the conditional elimination rule:
- A -> (B -> C)
- B
- Therefore, C
6. From 2 and 5, use the modus ponens rule:
- (C ^ D) -> E
- C
- Therefore, E
7. From 6 and 3, use modus ponens and conjunction elimination:
- -F -> (D ^ -E)
- (D ^ -E)
- Therefore, -F
8. From 7 and 4, use the modus ponens rule:
- -F
- Therefore, F
Since we have proven F under the assumption of B, we can conclude B -> F. And since we assumed A is true, we can also conclude A -> (B -> F).
Moving on to problem B:
1. (A v B) -> -(C ^ D)
2. (-C v -D) -> (E <-> F)
3. (E <-> F) -> (G ^ H)
To prove: (A v B) -> (H ^ G)
Again, to solve this problem, we'll use a conditional proof.
1. Assume A v B is true.
2. To prove: H ^ G
3. Assume H is true (under the assumption of A v B).
4. To prove: G
5. From 3 and 1, use the disjunction elimination rule:
- A v B
- Assume A:
- To prove G
- Assume B:
- To prove G
6. From 2 and 5, use the modus ponens rule:
- (E <-> F) -> (G ^ H)
- G ^ H
- Therefore, G
7. From 6 and 3, use conjunction elimination:
- G ^ H
- Therefore, H ^ G
Since we have proven H ^ G under the assumption of H, we can conclude that (A v B) -> (H ^ G).
I hope this helps! Let me know if you have any further questions.