The Principle of Duality states that starting with a Boolean expression, another Boolean expression can be obtained by :
1. Changing OR to AND
2. Changing AND to OR
3. Changing 0 to 1
4. Changing 1 to 0
My doubt is : Do we also complement any variable present in the expression?
For example, consider the expression:$$X+1=0$$
What will be the Dual for this?
Best Answer
No, we don't, because variables are self-dual. If we have a Boolean function $f(x_1, \dots, x_n)$ then the dual function $f^*(x_1, \dots, x_n)$ is defined to be $\overline{f}(\overline{x}_1,\dots, \overline{x}_n)$. In case $f(x_1, \dots, x_n) = x_i$ we have $$f^*(x_1, \dots, x_n) = \overline{\overline{x}}_i = x_i.$$
The Duality principle in general states that if you change all Boolean connectives in the expression to their duals (e.g., OR to AND and so on) then you will obtain the dual expression. Given that the variables are self-dual, you need to change them to themselves.
If $+$ represents OR then the dual is $X\cdot 0 = 1$, that is $0 = 1$, which is false along with your initial expression $X + 1 = 0$ equivalent to $1 = 0$. If $+$ represents sum modulo $2$ then the dual is $X +^* 0 = 1$, that is $\overline{X} = 1$. Here $X+^*Y = \overline{\overline{X} + \overline{Y}} = \overline{X + Y}$ is the dual for $X+ Y$.