# Prepositional Logic – Duality

## Introduction

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 with 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 validates this claim.  We can check this for other prepositional statements also.