Boolean Algebra Definition/Meaning:
An algebra that is particularly important in
computing. Formally it is a complemented distributive lattice. In a Boolean algebra there is a
set of elements B that consists of only 0 and 1. Further there will be two
binary operations, usually denoted by Ù and Ú (or by. and +) and called and and
or respectively. There is also a unary operation, denoted here by, and known
as the complement operation. These operations satisfy a series of laws, given in
the table, where x, y, and z denote arbitrary elements of B.
There are two very
common examples of Boolean algebras. The first consists of the set
B = {FALSE, TRUE}
with the binary AND and OR operations replacing Ù and
Ú
respectively, and the NOT operation producing complements. Thus 1 and 0 are
just TRUE and FALSE respectively. This idea can be readily extended to the set
of all n-tuples
(x1, x2, ....xn)
where each x1 is in B. The AND and OR operations are then extended to operate
between corresponding pairs of elements in each n-tuple to produce another n-tuple;
the NOT operation negates each item of an n-tuple.
The second common example of
a Boolean algebra is the set of subsets of a given set S, with the operations of'
intersection and union replacing
Ù and
Ú respectively; set complement fills
the role of Boolean algebra complement.
Boolean algebras, named for George Boole,
the 19th-century English mathematician, are fundamental to many aspects of
computing - switching theory, logic design, logic itself, and aspects of
algorithm design.
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