Some Symbolic Logic

Hi, I believe the problem is incorrect, and that you meant to write $(\sim B) \vee (\sim C)$ which is equivalent to $C \implies \sim B$ .

As it stands, the last proposition means that if $D$ is true then $A$ and $C$ are both true, and by the first statement, so is $B$ , and this is a consistent assignment of truth values, in that it does not contradict the second statement $\sim B \vee C$ which is true since $C$ is.

On the other hand if the second statement said $C \implies \sim B$ then the truth of $D$ would contradict this, so $D$ would have to be false.

Using the rules of inference from classical logic:

1) $A \equiv B$

2) $A \supset B$ (Biconditional elimination from (1))

3) $\sim B \lor C$

4) $B \supset C$ (Material implication from (3))

5) $A \supset C$ (Hypothetical syllogism from (2) and (4))

6) $D \supset (A \& \sim C)$

7) $(D \supset A) \& ( D \supset \sim C)$ (Distribution of (6))

8) $D \supset \sim C$ (Conjunctive elimination of (7))

9) $D \supset A$ (Conjunctive elimination of (7))

10) $D \supset C$ (Hypothetical syllogism of (9) and (5))

Assume indirect proof: Assume D to be true

11) $\sim C$ (Modus ponens of (8))

12) $C$ (Modus ponens of (10))

13) $C \& \sim C$ (Conjunctive introduction of (11) and (12))

(13) is a contradiction, hence the assumption that D is true is false.

C) D cannot be true

Since $A \Leftrightarrow B$ , we can infer, from the second premise, that $\neg A \vee C$ , which is the opposite of $A \wedge \neg C$ , which is therefore false.

The third premise then implies that D implies a falsehood, therefore D is false.

I think you made a second (besides the one pointed by Varsha Dani) mistake while defining D; you used set symbols (and that to wrongly as otherwise we would have no idea of contradicting D as it could include any possibility, besides having to include (A&C) in addition to something else) instead of logic symbols. There is technically no solution to this problem as I pointed out, but I assumed your mistake (luckily correctly) and got the right answer.