Boolean OR inverses

If we OR together value x and its inverse (NOT x, or \overline{x}), we get 1, because:

• 0 + 1 = 1
• 1 + 0 = 1
• So x + \overline{x} = 1

That’s because OR requires at least one side to be true if we want a true result, and we know that one of the sides is true and one is false, so at least one side is true.

A + \overline{A} = 1

Simplifying, just replace anywhere we have something ORed with its inverse with just 1. For example, \overline{A} + A simplifies to just 1.