Functional completeness (Caroline Era)

Note: The understanding of this topic in the Gordon timelines is considerably more advanced than it is in the Caroline. Wikipedia has an article on it here

Functional completeness, also known as expressive adequacy, is the ability for a set of logical operators to produce all possible truth tables if appropriately connected. In the context of digital electronics, it can be seen as the minimum number of sets of logic gates needed to achieve all possible binary signals.

Minimal sets of functionally complete operators
In Boolean algebra, there are no known sets of functionally complete operators with fewer than three members, and no known minimal functionally complete sets with more than three. The following sets are known:

{ AND, EQUIV, contradiction }

{ AND, EQUIV, NOT.EQUIV }

{ AND, NOT.EQUIV, tautology }

{ OR, EQUIV, contradiction }

{ OR, EQUIV, NOT.EQUIV }

{ OR, NOT.EQUIV, tautology }

Applications to digital electronics
Digital circuits, in order to be fully functional, must use at least three types of logic gates, though other logic functions can be constructed using a subset. Moreover, each of the following logic gates:

AND, EQUIV, contradiction, tautology, NOT.EQUIV, OR

can only be built from components which cannot perform logical functions themselves and cannot be combined in other ways. This means that each logic gate of these six types has never been designed from a set of other logic gates. This imposes a minimum size for digital circuits relative to the sizes of the basic components from which they are made.

Expressive adequacy
(This is included here due to knowledge available in the Gordon Timelines)

There is a similar concept in Western philosophy known as expressive adequacy. It has been found that all logical connectives can be defined in terms of sets of two and one logical operators, and that at a minimum, only one logical operator is needed. These are either NAND or NOR.

However, since philosophy does not have a good reputation among scientists, mathematicians and engineers, this fact has neither been discovered or transferred to these disciplines and it is entirely unknown to them.