# Boolean algebra

Boolean algebra is algebra for binary (0 means false and 1 means true). It uses normal maths symbols, but it does not work in the same way. It is named for George Boole, who invented it in the middle 19th century. In the 20th century boolean algebra came to be much used for logic gates.

## NOT gate

NOT
0 1
1 0

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

${\bar {1}}=0$ ${\bar {0}}=1$ ${\bar {\mbox{A}}}={\mbox{Q}}$ It means the output is not the input.

## AND gate

AND 0 1
0 0 0
1 0 1

The AND operator is written as $\cdot$ like this:

$0\cdot 0=0$ $0\cdot 1=0$ $1\cdot 0=0$ $1\cdot 1=1$ The output is true only if one and the other input is true.

## OR gate

OR 0 1
0 0 1
1 1 1

The OR operator is written as $+$ like this:

$0+0=0$ $0+1=1$ $1+0=1$ $1+1=1$ One or the other input can be true for the output to be true.

## XOR gate

XOR 0 1
0 0 1
1 1 0

XOR basically means "exclusive or", meaning one input or the other must be true, but not both. It is also sometimes called NOR, which means the same thing.

The XOR operator is written as $-$ like this:

$0-0=0$ $0-1=1$ $1-0=1$ $1-1=0$ To make it more simple, one or the other input must be true, but not both.

## Identities

Different gates can be put together in different orders:

${\overline {{\mbox{A}}\cdot {\mbox{B}}}}$ 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 like this: ${\overline {\mbox{A}}}\cdot {\overline {\mbox{B}}}$ ${\mbox{A}}+1=1$ ${\mbox{A}}\cdot 1={\mbox{A}}$ which is called XOR identity table

XOR 1 0 Any
1 TRUE 0 0
0 0 0 ${\overline {ANY}}$ Any 0 ${\overline {ANY}}$ $\{Any\}$ , if $ANY=\{x|\{x\}=\{\{TRUE\}\lor \{{\overline {TRUE}}\},\};\land (TRUE,0)\vdash TRUE\land {\overline {0}}=\{x\}$ .[source?]

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

## DeMorgan's laws

Augustus De Morgan found out that it is possible to change a $+$ sign to a $\cdot$ sign and make or break a bar. See the 2 examples below:

${\overline {{\mbox{A}}+{\mbox{B}}}}={\overline {\mbox{A}}}\cdot {\overline {\mbox{B}}}$ ${\overline {{\mbox{A}}\cdot {\mbox{B}}}}={\overline {\mbox{A}}}+{\overline {\mbox{B}}}$ "Make/break the bar and change the sign."