Question

The negation of the statement \(((A\land(B\lor C)) ⇒ (A\lor B)) ⇒ A\) is

• A. a fallacy
• B. equivalent to \(B\lor \sim C\)
• C. equivalent to \(\sim A\)
• D. equivalent to \(\sim C\)

Answer 1

The Correct Option is C

To find the negation of the given proposition \(((A \land (B \lor C)) \Rightarrow (A \lor B)) \Rightarrow A\), we will first express each part using logical equivalences and then determine its negation.

1. The main statement is of the form \(P \Rightarrow A\), where \(P\) is \((A \land (B \lor C)) \Rightarrow (A \lor B)\).

2. The implication \(P \Rightarrow A\) can be rewritten as \(\sim P \lor A\) using the equivalence \(P \Rightarrow Q \equiv \sim P \lor Q\).

3. The sub-expression \(P = (A \land (B \lor C)) \Rightarrow (A \lor B)\) can be rewritten using the same equivalence: \(\sim (A \land (B \lor C)) \lor (A \lor B)\).

   • The expression \(\sim (A \land (B \lor C))\) can be simplified using De Morgan’s laws: \(\sim A \lor \sim (B \lor C)\), which further simplifies to \(\sim A \lor (\sim B \land \sim C)\).

4. Substituting back into the expression for \(P\), we have:

   \((\sim A \lor (\sim B \land \sim C)) \lor (A \lor B)\).

5. Now, substitute this expression for \(P\) back into the overall statement:

   \(\sim ((\sim A \lor (\sim B \land \sim C)) \lor (A \or B)) \lor A\).

6. To find the negation of the entire statement, apply the negation directly:

   \(\sim (\sim ((\sim A \lor (\sim B \land \sim C)) \lor (A \lor B)) \lor A)\).

7. Using the equivalence \(\sim (P \lor Q) \equiv \sim P \land \sim Q\), the negation simplifies to:

   \((\sim A \lor (\sim B \land \sim C)) \lor (A \or B)\) and \(\sim A\).

8. Thus, the negation of the original statement simplifies to \(\sim A\).

Therefore, the negation of the given statement is equivalent to \(\sim A\), confirming that the correct answer is "equivalent to \(\sim A\)".