Boolean algebra

In mathematics, Boolean algebra is an algebra for binary digits (where 0 means false and 1 means true).^[1] It is equipped with three operators: conjunction (AND), disjunction (OR) and negation (NOT).^[2] It uses normal math symbols, but it does not work in the same way. It is named for George Boole,^[3] who invented it in the middle 19th century. Boolean algebra did not get much attention except from mathematicians until the 20th century when engineers began using it for logic gates.

NOT gate

Main article: NOT gate

NOT
0	1
1	0

^[4]

The NOT operator is written with a bar over numbers or letters like this:

${\displaystyle {\bar {1}}=0}$

${\displaystyle {\bar {0}}=1}$

${\displaystyle {\bar {\mbox{A}}}={\mbox{Q}}}$

It means that the output is not the input.

AND gate

Main article: AND gate

AND	0	1
0	0	0
1	0	1

^[4]

The AND operator is written as ${\displaystyle \cdot }$ like this:^[5]

${\displaystyle 0\cdot 0=0}$

${\displaystyle 0\cdot 1=0}$

${\displaystyle 1\cdot 0=0}$

${\displaystyle 1\cdot 1=1}$

The output is true if and only if one and the other input is true.

OR gate

Main article: OR gate

OR	0	1
0	0	1
1	1	1

^[4]

The OR operator is written as ${\displaystyle +}$ like this:^[5]

${\displaystyle 0+0=0}$

${\displaystyle 0+1=1}$

${\displaystyle 1+0=1}$

${\displaystyle 1+1=1}$

If one or the other input is true, then the output to be true (and false otherwise).

XOR gate

Main article: XOR gate

XOR	0	1
0	0	1
1	1	0

^[4]

XOR basically means "exclusive or", meaning one input or the other must be true, but not both.

The XOR operator is written as ${\displaystyle -}$ like this:

${\displaystyle 0-0=0}$

${\displaystyle 0-1=1}$

${\displaystyle 1-0=1}$

${\displaystyle 1-1=0}$

In other words, the XOR operator returns true precisely when one or the other input is true—but not both.

Identities

Different gates can be put together in different orders:

${\displaystyle {\overline {{\mbox{A}}\cdot {\mbox{B}}}}}$ is the same as an AND then a NOT. This is called a NAND gate.

It is not the same as a NOT then an AND: ${\displaystyle {\overline {\mbox{A}}}\cdot {\overline {\mbox{B}}}}$

${\displaystyle {\mbox{A}}+1=1}$

${\displaystyle {\mbox{A}}\cdot 1={\mbox{A}}}$

which is called AND identity table

AND	1	0	Any
1	TRUE	0	0
0	0	0	${\displaystyle {\overline {ANY}}}$
Any	0	${\displaystyle {\overline {ANY}}}$	${\displaystyle \{Any\}}$

, if ${\displaystyle ANY=\{x|\{x\}=\{\{TRUE\}\lor \{{\overline {TRUE}}\},\};\land (TRUE,0)\vdash TRUE\land {\overline {0}}=\{x\}}$.^[source?]

or if ${\displaystyle ANY=\{x\|\{TRUE\},\{{\overline {TRUE}}\}.\},}$=TRUE, TRUE.,

De Morgan's laws

Augustus De Morgan discovered that it is possible to preserve the truth values of Boolean expressions by changing a ${\displaystyle +}$ sign to a ${\displaystyle \cdot }$ sign, while making or breaking a bar. That is:

${\displaystyle {\overline {{\mbox{A}}+{\mbox{B}}}}={\overline {\mbox{A}}}\cdot {\overline {\mbox{B}}}}$

${\displaystyle {\overline {{\mbox{A}}\cdot {\mbox{B}}}}={\overline {\mbox{A}}}+{\overline {\mbox{B}}}}$

These findings are commonly known as De Morgan's laws.

Related pages

• Abstract algebra
• Boolean data type
• Boolean satisfiability problem
• Fuzzy logic
• Propositional logic