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

To find the Dual of the given statement:

Given statement:
p ∧ [ ¬q ∨ (p ∧ q) ∨ ¬r ]

Step 1: Understand the Concept of Duality
The dual of a logical statement is obtained by interchanging:

∧ (AND) with ∨ (OR)
∨ (OR) with ∧ (AND)

Negations (¬) and variables remain unchanged.

Step 2: Apply Dual Transformation
Original Expression:
p ∧ [ ¬q ∨ (p ∧ q) ∨ ¬r ]

Replace every ∧ with ∨, and every ∨ with ∧:

p ∨ [ ¬q ∧ (p ∨ q) ∧ ¬r ]

Final Answer (Dual Statement):
p ∨ [ ¬q ∧ (p ∨ q) ∧ ¬r ]