Mathematical Logic: Duality

Mathematical Logic

Duality

Definition 12.19

The dual of a statement formula is obtained by replacing ∨ by ∧ , ∧ by ∨ , T by F F by T. A dual is obtained by replacing T (tautology) by (contradiction), F and, by T.

Remarks

(1) The symbol ¬ is not changed while finding the dual.

(2) Dual of a dual is the statement itself.

(3) The special statements T (tautology) and F (contradiction) are duals of each other.

(4) T is changed to F and vice-versa.

Principle of Duality

If a compound statement S₁ contains only ¬, ∧ , and ∨ and statement S₂ arises from S₁ by replacing ∧ by ∨ , and, ∨ by ∧ then S₁ is a tautology if and only if S₂ is a contradiction.

For example

(i) The dual of ( p ∨ q) ∧ (r ∧ s ) ∨ F is ( p ∧ q) ∨ ( r ∨ s ) ∧ T.

(ii) The dual of p ∧ [¬ q ∨ ( p ∧ q) ∨ ¬ r ] is p ∨ [¬ q ∧ ( p ∨ q) ∧ ¬ r ].