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[change | edit source]

NOT
0 1
1 0

[2]

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[change | edit source]

AND 0 1
0 0 0
1 0 1

[2]

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[change | edit source]

OR 0 1
0 0 1
1 1 1

[2]

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[change | edit source]

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 - like this:

0 - 0 = 0
0 - 1 = 1
1 - 0 = 1
1 - 1 = 0

Identities[change | edit source]

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\}.[source?]


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

DeMorgan's laws[change | edit source]

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."

Other pages[change | edit source]

References[change | edit source]