201 about logic

Item 1. You can have a fine time proving DeMorgan's Laws with Venn diagrams.

Item 2. We saw three logical functions of two variables: NOT, AND, and OR. There are in fact sixteen such functions in all. Why?

These three are universal, which means that you can find an equivalent for any of the other thirteen functions using AND, OR, and NOT.

Also, NOT and AND are universal, so you can make an OR out of just NOTs and ANDs. And you can make an AND out of NOTs and ORs.

One of the sixteen is called NAND: p NAND q is the NOT of p AND q. NAND is universal all by itself. NOR is also universal. Integrated logic circuitry is generally made of either NANDs or NORs rather than combinations of AND, OR, and NOT, because it's cheaper to make just one kind of logic gate.