Prepositional Logic – Duality

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We know that the simple statements are represented as p, q and so on. Suppose we are given a compound preposition

p ∧  q

There is another property of compound prepositions called the duality. The dual of the above statement is

p ∨ q

Therefore, the dual can be achieved by interchanging ∧ by  ∨ or interchanging  ∨ by ∧. This means ∧ and ∨ are dual of each other.

Dual for another type of statement

We will see an example of another type of prepositional statement in which we use True or False directly instead of a variable.

For example,

p ∨ T  is a compound preposition which True as one of the variable.

The dual of such a statement can be obtained by interchanging ∨ by ∧ and interchanging True with False.

Each of the statement can be derived from the other.

Understanding Duality using a Truth Table

These are the truth tables for statements (p ∧  q ) and (p ∨ q).

If we interchange the  (p ∧  q ) by (p ∨ q).or (p ∨ q). by (p ∧  q ) , we get the dual of each other. The truth table validate this claim.  We can check this for other prepositional statement also.

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