Boolean functions: functional completeness

Sum of products can represent any truth table:

z = p1 + p2 . . . + pn

each pi = l1l2 . . . lm

and each lk is a literal

Applying double negation to the right hand side,

z = ~~(p1 + p2 . . . + pn)

= ~(~p1 * ~p2 . . . * ~pn)

by DeMorgan's law

OR has been eliminated.

Therefore, {NOT, AND} is a functionally complete set.

Similarly, {NOT, OR} is also functionally complete.

Are {AND} and {OR} functionally complete?  No.

Consider any Boolean function composed of only these functions.

If all of the inputs are 1, then the output MUST be 1,

and if all the inputs are 0, then the output MUST be 0.

1 AND 1 == 1, 1 OR 1 == 1

0 AND 0 == 0, 0 OR 0 == 0

However, it is certainly possible to contruct a truth table where the output

is 0 when all the inputs are 1, and vice versa.