Question

Write the dual of the statement \( p \land [\neg q \lor (p \land q) \lor \neg r] \).

Hint:

To find the dual quickly, just swap \( \land \leftrightarrow \lor \) everywhere while keeping negations the same.

Answer 1

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 \[ \boxed{p \lor [\neg q \land (p \lor q) \land \neg r]} \]