Boolean AND inverses

If we AND together value x and its inverse (NOT x, or \overline{x}), we get 0, because:

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

This is because the AND requires both sides to be true if we want a true result, but we already know one of the sides is false and one is true, so they’re NOT both true.

A \cdot \overline{A} = 0

We can replace anywhere we have something ANDed with its inverse with just 0 in order to simplify. For example, A \cdot \overline{A} simplifies to just 0.