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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:

• Material conditional: ($x \to y = \lnot x \lor y$)
• Exclusive disjuction: ($x \oplus y = \lnot(x \leftrightarrow y \land y \leftrightarrow x)$)
• Bicondional: ($x \leftrightarrow y = \lnot (x \oplus y)$)

## 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.