Boolean AND inverses

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

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.

flashcards

QuestionAnswer
What is the result of ANDing a Boolean value with its inverse?A \cdot \overline{A} = 0, because for both A=0 (0 \cdot 1 = 0) and A=1 (1 \cdot 0 = 0) the result is 0.
How can you simplify an expression containing a variable ANDed with its own inverse?You can replace the term with 0, e.g., A \cdot \overline{A} simplifies to 0.
Why does x \cdot \overline{x} always equal 0?Because AND requires both inputs to be true for a true output, but one side is always false and the other is always true, so they are never both true.