A Boolean algebra is a structure (`B`

, `and`

, `or`

, `not`

, `0`

, `1`

) such that,
`B`

is a non-empty set, `and`

and `or`

are binary functions from `B`

x`B`

to `B`

,
`not`

is a unary function from `B`

to `B`

, and both `0`

and `1`

are constants which are members of the set `B`

.
The following laws hold:

- for all
`a`

in`B`

:`a and a = a`

-*idempotency*. - for all
`a`

in`B`

:`a or a = a`

-*idempotency*. - for all
`a`

,`b`

in`B`

:`a and b = b and a`

-*commutativity*. - for all
`a`

,`b`

in`B`

:`a or b = b or a`

-*commutativity*. - for all
`a`

,`b`

,`c`

in`B`

:`a and (b and c) = (a and b) and c`

-*associativity*. - for all
`a`

,`b`

,`c`

in`B`

:`a or (b or c) = (a or b) or c`

-*associativity*. - for all
`a`

,`b`

,`c`

in`B`

:`a and (b or c) = (a and b) or (a and c)`

-*distributivity*. - for all
`a`

,`b`

,`c`

in`B`

:`a or (b and c) = (a or b) and (a or c)`

-*distributivity*. - for all
`a`

in`B`

:`a and 1 = a`

-*verum*. - for all
`a`

in`B`

:`a or 1 = 1`

-*verum*. - for all
`a`

in`B`

:`a and 0 = 0`

-*falsum*. - for all
`a`

in`B`

:`a or 0 = a`

-*falsum*.

We can use digital voltage levels and logic gates as the interpretations of the elements of Boolean algebra as follows:

`B`

is the set {low voltage level, high voltage level}.`and`

is the AND gate.`or`

is the OR gate.`not`

is the NOT gate.`0`

is the low voltage level.`1`

is the high voltage level.

It can be shown by the logic gate truth tables that this interpretation satisfies the laws of Boolean algebra.

Copyright © 2014 Barry Watson. All rights reserved.