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# Mathematical Logic: Duality

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.

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 ].

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