(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:

        NOT A  =  A NOR A
        A OR B  =  NOT (A NOR B)

and DeMorgan's Theorem.