<logic> (Commonly known as "Boolean algebra") A mathematical
system concerning the two truth values, TRUE and FALSE and
the functions AND, OR, NOT. Two-valued logic is one of
the cornerstones of logic and is also fundamental in the
design of digital electronics and programming languages.
The term "Boolean" is used here with its common meaning -
two-valued, though strictly Boolean algebra is more general
than this.
Boolean functions are usually represented by truth tables
where "0" represents "false" and "1" represents "true". E.g.:
A | B | A AND B
--+---+--------
0 | 0 | 0
0 | 1 | 0
1 | 0 | 0
1 | 1 | 1
This can be given more compactly using "x" to mean "don't
care" (either true or false):
A | B | A AND B
--+---+--------
0 | x | 0
x | 0 | 0
1 | 1 | 1
Similarly:
A | NOT A A | B | A OR B
--+------ --+---+--------
0 | 1 0 | 0 | 0
1 | 0 x | 1 | 1
1 | x | 1
Other functions such as XOR, NAND, NOR or functions of
more than two inputs can be constructed using combinations of
AND, OR, and NOT. AND and OR can be constructed from each
other using DeMorgan's Theorem:
A OR B = NOT ((NOT A) AND (NOT B))
A AND B = NOT ((NOT A) OR (NOT B))
In fact any Boolean function can be constructed using just NOR
or just NAND using the identities:
