Prepositional Logic – Duality

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.

pqp \land q
TTT
TFF
FTF
FFF
Truth table for “and”
pqp \lor q
TTT
TFT
FTT
FFF
Truth table for “or”

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.

Please support us by disabling your adblocker or whitelist this site from your adblocker. Thanks!

turn of adblocker imag