# Boolean logic: why is the "opposite" of and equal to or?

In boolean logic, why is the "opposite" of AND (&) equal to OR (|)?

For example, why would !(A & B) be equal to (!A | !B)?

I understand that not A would simply be !A, but why does AND change to OR?

This is called DeMorgan's law, and let us think about it like this... By the way, let

• $$\land$$ be &&
• $$\lor$$ be ||
• $$\neg$$ be !

You have a door, and the door has two locks. To open the door you therefore need keys $$A$$ and $$B$$, written $$A \land B$$.

Well, suppose that someone is not able to open the door. Well, then that must be because they don't have both keys, or, $$\neg (A \land B)$$.

But analysing this situation, if you do not have two keys, you either lack $$A$$ or you lack $$B$$, or in other words, you either $$\neg A$$ or you $$\neg B$$.

Hence, $$\neg (A \land B) = \neg A \lor \neg B$$.

You can go through the exercise with a door where you need 1 of 2 keys to open for the reverse case.