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.

^[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.

^[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.

^[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).

^[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.

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.,

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.

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

• Boolean algebra on All About Circuits
• Boolean algebra Citizendium

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