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Boolean algebra is a branch of algebra concerned with variables whose values are truth values, true and false, usually denoted with the integers 0 and 1 respectively. The set of these truth values is known as a boolean domain. The Boolean domain acts the same way as the two-element set Integer Modulo 2. Boolean algebra is named after George Boole.

The basic operations of Boolean algebra are Conjunction (\land), Disjunction (\lor), and Negation (\lnot). The operations can be defined as:

x \land y = xy
x \lor y = x+y-(xy)
\lnot x = 1-x

where x,y\in\{0,1\}.

They can also be defined using a table called a truth table:

x y x \land y
1 1 1
1 0 0
0 1 0
0 0 0
x y x \lor y
1 1 1
1 0 1
0 1 1
0 0 0
x

\lnot x

1 0
0 1

OperationsEdit

The operations on the boolean domain are known as boolean-valued operations. These include conjunction, disjunction, and negation. Here are more:

Boolean lawsEdit

Law Name: Law:
Associativity of disjunction x \lor (y \lor z)=(x \lor y) \lor z
Associativity of conjunction

x \land (y \land z)=(x \land y) \land z

Commutativity of disjunction x \lor y = y \lor x
Commutativity of conjunction x \land y = y \land x
Distributivity of conjunction over disjunction x \land (y \lor z) = (x \land y)\lor(z \land z)
Identity for disjunction x \lor 1 = x
Identity for conjunction x \land 0 = x
Annihilator for disjunction x \land 0 = 0
Annihilator for conjunction x \lor 1 = 1
Idempotence of disjunction x \lor x = x
Idempotence of conjunction x \land x = x
Absorption 1 x \land (x \lor y) = x
Absorption 2 x \lor (x \land y) = x
Complementation 1: x \land \lnot x =0
Complementation 2: x \lor \lnot x = 1
Double negation \lnot (\lnot x) = x
DeMorgan's laws
  • \lnot x \land \lnot y = \lnot (x \lor y)
  • \lnot x \lor \lnot y = \lnot (x \land y)
Distributivity of disjunction over conjunction x \lor ( y \land z) = (x \lor y) \land (x \lor z)

ApplicationsEdit

LogicEdit

Boolean algebra is often used to define the semantics of several formal systems, such as classical propositional logic . Many concepts of syntax in those formal systems come from Boolean logic.  These logic systems are referred to as Boolean logic

ComputingEdit

Boolean logic is often used for logic gates in computers.

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