Search code examples
boolean-operations

Applying the dual/duality principle to Boolean expressions

First off, my textbook says the Dual Principle is switching ORs with ANDs and ANDs with ORs, but leaving the variables themselves in their complemented or uncomplemented form. This differs from what I've read online about the Duality Principle, which states that you must interchange the ORs and ANDs as well as complement each individual variable. Regardless, these are just two different definitions. My question lies in how to apply the Dual and/or Duality Principle to the following situation:

(xy)'

Would the dual (according to the Duality Principle) be 

(x' + y')'?

In other words, does the outside NOT stay there? Or, are all NOTs themselves, even if they are "outer NOTs" complemented? This would lead to: 

(x' + y').

I'm pretty sure it is the latter.

Solution

  • It is like this:

     (x' + y')' = (x')'(y')' = xy

    The rules applied here are:

     (x + y)' = x'y', and
 (x')' = x