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Boolean algebra




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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 after its creator George Boole.[1]

NOT gate

NOT
0 1
1 0
[2]

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

[math]\bar{1} = 0[/math]
[math]\bar{0} = 1[/math]
[math]\bar{\mbox{A}} = \mbox{Q}[/math]

It means the output is not the input.

AND gate

AND 0 1
0 0 0
1 0 1
[2]

The AND operator is written as [math]\cdot[/math] like this:

[math]0 \cdot 0 = 0[/math]
[math]0 \cdot 1 = 0[/math]
[math]1 \cdot 0 = 0[/math]
[math]1 \cdot 1 = 1[/math]

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

OR gate

OR 0 1
0 0 1
1 1 1
[2]

The OR operator is written as [math]+[/math] like this:

[math]0 + 0 = 0[/math]
[math]0 + 1 = 1[/math]
[math]1 + 0 = 1[/math]
[math]1 + 1 = 1[/math]

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
[2]

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

The XOR operator is written as [math]-[/math] like this:

[math]0 - 0 = 0[/math]
[math]0 - 1 = 1[/math]
[math]1 - 0 = 1[/math]
[math]1 - 1 = 0[/math]

Identities

Different gates can be put together in different orders:

[math]\overline{\mbox{A} \cdot \mbox{B}}[/math] 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: [math]\overline{\mbox{A}} \cdot \overline{\mbox{B}}[/math]

[math]\mbox{A} + 1 = 1[/math]
[math]\mbox{A} \cdot 1 = \mbox{A}[/math]

which is called XOR identity table

XOR 1 0 Any
1 TRUE 0 0
0 0 0 [math]\overline{ANY}[/math]
Any 0 [math]\overline{ANY}[/math] [math]\{Any\}[/math]

, if [math] ANY=\{x|\{x\}=\{\{TRUE\}\or\{\overline{TRUE}\}, \};\and (TRUE, 0) \vdash TRUE \and \overline{0} = \{x\}[/math].[source?]


or if [math] ANY=\{x \|\{TRUE\}, \{\overline{TRUE}\} .\},[/math]=TRUE, TRUE.,

DeMorgan's laws

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

[math]\overline{\mbox{A} + \mbox{B}} = \overline{\mbox {A}} \cdot \overline{\mbox{B}}[/math]
[math]\overline{\mbox{A} \cdot \mbox{B}} = \overline{\mbox {A}} + \overline{\mbox{B}}[/math]
[math]\overline{\mbox{A} \cdot \mbox{B}} = \overline{\mbox {A}} + \overline{\mbox{B}} = [/math][math] \overline{ {\overline{\mbox{A}} \cdot \overline{\mbox{B}} } } = NOT (\overline{\mbox{A}} \cdot \overline{\mbox{B}})[/math]


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

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References


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