Involution law

Also called: double negation, double complement, involution, double complement law, double negation law

The Boolean law A'' = A: complementing a value twice gives back the original, so two bars over the same thing cancel.

The involution law says two NOTs cancel:

= A

Flip 0 and you get 1; flip again and you are back to 0. The same holds starting from 1. Typed, that is A'' = A.

It applies to whole expressions too: = . In a circuit, two inverters in a row act like a plain wire.

Where it shows up:

After de morgans laws. Breaking a bar over gives , and the involution law turns into B.
Bubble pushing. Two bubbles on the same wire cancel, which is how an AND-OR circuit becomes a NAND-NAND circuit. See bubble pushing.
Odd numbers of bars. Three bars equal one: = .


0	1	0
1	0	1

Worked example

Example

Remove the bar from .

Cancelling bars of different lengths. In the long bar and the short bars are not on the same thing, so use De Morgan first: the result is .
Losing count with many bars. An even number cancels completely; an odd number leaves one.

Interactive questions with instant feedback and a worked solution for every wrong answer.

Involution law is taught in Boolean Algebra.