# 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.

$p \land q$

There is another property of compound prepositions called the duality; Therefore, the dual of above statement is

$p \lor q$

The dual can be achieved by interchanging $\land$ by $\lor$ or interchanging $\lor$ by $\land$. This means $\land$ and $\lor$ is 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,

$p \lor T$

is a preposition with T or True as one of the variable.

The dual of such a statement can be obtained by interchanging $\lor$ by $\land$ and interchanging True with False or False with True.

Each of the statement can be derived from the other.

### Understanding Duality Using Truth Tables

These are the truth tables for the statements – $p \land q$ and $p \lor q$.

If we change the operator for $p \land q$ with $p \lor q$, or vice-versa, we get dual of each other. The truth table validates this claim. We can check this for other prepositional statements also.