Question:
Write the dual of the statement \( p \land [\neg q \lor (p \land q) \lor \neg r] \).
Write the dual of the statement \( p \land [\neg q \lor (p \land q) \lor \neg r] \).
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To find the dual quickly, just swap \( \land \leftrightarrow \lor \) everywhere while keeping negations the same.
Updated On: Feb 25, 2026
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Solution and Explanation
Concept:
In propositional logic, the dual of a statement is obtained by:
Replacing every \( \land \) (AND) with \( \lor \) (OR)
Replacing every \( \lor \) (OR) with \( \land \) (AND)
Negations remain unchanged
Constants (if present): \(0 \leftrightarrow 1\)
No structural rearrangement is required—only operator swapping.
Step 1: Given expression \[ p \land [\neg q \lor (p \land q) \lor \neg r] \]
Step 2: Replace outer AND with OR \[ p \lor [\neg q \lor (p \land q) \lor \neg r] \]
Step 3: Convert inner ORs to ANDs and AND to OR \[ p \lor [\neg q \land (p \lor q) \land \neg r] \] Final Answer: The dual is \[ \boxed{p \lor [\neg q \land (p \lor q) \land \neg r]} \]
Replacing every \( \land \) (AND) with \( \lor \) (OR)
Replacing every \( \lor \) (OR) with \( \land \) (AND)
Negations remain unchanged
Constants (if present): \(0 \leftrightarrow 1\)
No structural rearrangement is required—only operator swapping.
Step 1: Given expression \[ p \land [\neg q \lor (p \land q) \lor \neg r] \]
Step 2: Replace outer AND with OR \[ p \lor [\neg q \lor (p \land q) \lor \neg r] \]
Step 3: Convert inner ORs to ANDs and AND to OR \[ p \lor [\neg q \land (p \lor q) \land \neg r] \] Final Answer: The dual is \[ \boxed{p \lor [\neg q \land (p \lor q) \land \neg r]} \]
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