Boolean functions

There are 16 possible functions with 2 bits of input and 1 bit of output.

Of these, only 6 are gates:

AND, OR, XOR, NAND, NOR, XNOR

All possible Boolean functions can be written using at most 3 gates:

Set {AND, OR, NOT} is computationally complete.

Also {NAND}, {NOR}, and some others.

Example: use NAND to implement OR

x | y

= ~~(x | y)

= ~(~x & ~y)

= ~x NAND ~y

Looks like we also need NOT

However, consider the following:

~x

= ~x | ~x

a == a | a

= ~(x & x)

DeMorgan's law

= x NAND x

Definition of NAND

So,

x | y = (x NAND x) NAND (y NAND y)

A bit ugly, perhaps, but true.