F → (B ∨ A)D → F¬(B ∨ A)..∴ ¬(D ∧ C)Which of the following can be derived from the premises?¬B ∨ ¬A¬(D ∨ C)¬B ∧ ¬A¬D

The premises given are:

1. F → (B ∨ A)
2. D → F
3. ¬(B ∨ A)

We are asked to derive which of the following conclusions:

1. ¬B ∨ ¬A
2. ¬(D ∨ C)
3. ¬B ∧ ¬A
4. ¬D

Let's go step by step:

1. ¬B ∨ ¬A: This is the De Morgan's law equivalent of ¬(B ∨ A), which is one of the premises. So, this can be derived from the premises.

2. ¬(D ∨ C): There is no information given about C in the premises, so we cannot derive any conclusion that involves C.

3. ¬B ∧ ¬A: This is not equivalent to ¬(B ∨ A), so it cannot be derived from the premises.

4. ¬D: From the second premise (D → F), we know that if D is true, then F is true. But from the third premise, we know that F is false (because F → (B ∨ A) and ¬(B ∨ A) are given). So, D must also be false. Therefore, ¬D can be derived from the premises.

So, the conclusions that can be derived from the premises are ¬B ∨ ¬A and ¬D.