Show that A →→D follows logically from the premises.

ABVC, BA and D→ -C

To show that \( A \to\to D \) follows logically from the premises, we need to use the given information step-by-step and prove that if \( A \) holds, then \( D \) must hold. The premises are:

1. \( A \lor B \lor C \) \(\tag{1}\)
2. \( B \land A \) \(\tag{2}\)
3. \( D \to \neg C \) \(\tag{3}\)

### Step 1: Analyze \( B \land A \)

Premise (2) states \( B \land A \). This means both \( A \) and \( B \) are true. Therefore, we know for certain that \( A \) and \( B \) are true.

### Step 2: Use \( A \lor B \lor C \)

Premise (1) is \( A \lor B \lor C \), which says at least one of \( A \), \( B \), or \( C \) is true. Since we know from premise (2) that both \( A \) and \( B \) are true, this premise holds.

### Step 3: Analyze \( D \to \neg C \)

Premise (3) states \( D \to \neg C \), which means if \( D \) is true, then \( C \) must be false (i.e., \( \neg C \)). To logically derive this, consider the contrapositive form of this implication: \( C \to \neg D \). This means if \( C \) is true, then \( D \) must be false.

### Step 4: Consider the Case for \( C \)

We already know from \( A \land B \) that \( A \) and \( B \) are both true. Therefore, for \( A \lor B \lor C \) to be satisfied, \( C \) doesn't need to be true, since \( A \) and \( B \) are already true. So, let us consider the scenario where \( C \) is false.

If \( C \) is false, then the contrapositive form of premise (3) doesn't apply. We don't have any restriction that \( D \) must be false, so \( D \) can be true.

### Step 5: Conclusion

Since \( A \) is true (from premise 2), and if \( C \) is false, then there's no reason \( D \) cannot be true. Therefore, we can conclude that \( A \to\to D \), i.e., if \( A \) holds, \( D \) follows logically.

Thus, \( A \to\to D \) follows from the premises.

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Would you like further details or explanations?

Here are 5 related questions for further exploration:
1. How does the contrapositive of an implication work in logic?
2. What are the steps to prove implications in propositional logic?
3. Can \( D \) be false in any scenario under these premises?
4. How would the result change if \( D \to \neg C \) was \( D \leftrightarrow \neg C \)?
5. What is the difference between \( \to \) and \( \to\to \) in logic?

**Tip:** When dealing with logical implications, remember that the contrapositive \( p \to q \) is logically equivalent to \( \neg q \to \neg p \).

Show that A →→D follows logically from the premises: ABVC, BA, and D→ -C.

Explore a detailed step-by-step solution to prove that A →→ D follows logically from the premises ABVC, BA, and D→ -C. This problem focuses on key concepts such as implications, contraposition, and the application of logical theorems. Suitable for undergraduate students studying logic or discrete mathematics.

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