Boolean algebra

Boolean algebra is algebra for binary (0 meaning false, or 1 meaning true). It uses normal maths symbols, but it does not work in the same way. It is named after its creator George Boole.

NOT gate

Main page: 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

Main page: 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

Main page: 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

Main page: XOR gate

XOR	0	1
0	0	1
1	1	0

One or the other input can be true to make the output true, but NOT both.

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

$0 - 0 = 0$

$0 - 1 = 1$

$1 - 0 = 1$

$1 - 1 = 0$

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 $ANY=\{x|\{x\}=\{\{TRUE\}\or\{\overline{TRUE}\}, \};\and (TRUE, 0) \vdash TRUE \and \overline{0} = \{x\}$.

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:

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

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

$\overline{\mbox{A} \cdot \mbox{B}} = \overline{\mbox {A}} + \overline{\mbox{B}} =$$\overline{ {\overline{\mbox{A}} \cdot \overline{\mbox{B}} } } = NOT (\overline{\mbox{A}} \cdot \overline{\mbox{B}})$

"Make/break the bar and change the sign."

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