What does duality principle state? What is its usage in boolean algebra?

Q. What does duality principle state? What is its usage in boolean algebra?

Answer :-

This states that starting with a boolean relation, another boolean relation can be derived by:

1. changing each OR sign (+) to an AND sign (.)
2. changing each AND sign (.) to an OR sign (+)
3. replacing each 0 by 1 and each 1 by 0.

The derived relation using duality principle is called dual of original expression.

• The postulates and theorems which have been presented can all be divided into pairs.

For example:-

X+X.Y=X

Its dual will be

X.(X+Y)=X  (Remember change. to + and vice-versa; complement 0 and 1.)

Similarly,

(X+Y)+Z=X+(Y+Z) is the dual of (X. Y).Z = X. (Y.Z)

And X+0 = X is dual of X.1=X

In proving the theorems or rules of boolean algebra, it is then necessary to prove only one theorem, and the dual of the theorem follows necessarily. In effect, all boolean algebra is predicated on this two-for-one basis.