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 after its creator George Boole.[1]
NOT gate
| NOT | |
|---|---|
| 0 | 1 |
| 1 | 0 |
The NOT operator is written with a bar over numbers or letters like this:
- [math]\displaystyle{ \bar{1} = 0 }[/math]
- [math]\displaystyle{ \bar{0} = 1 }[/math]
- [math]\displaystyle{ \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 |
The AND operator is written as [math]\displaystyle{ \cdot }[/math] like this:
- [math]\displaystyle{ 0 \cdot 0 = 0 }[/math]
- [math]\displaystyle{ 0 \cdot 1 = 0 }[/math]
- [math]\displaystyle{ 1 \cdot 0 = 0 }[/math]
- [math]\displaystyle{ 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 |
The OR operator is written as [math]\displaystyle{ + }[/math] like this:
- [math]\displaystyle{ 0 + 0 = 0 }[/math]
- [math]\displaystyle{ 0 + 1 = 1 }[/math]
- [math]\displaystyle{ 1 + 0 = 1 }[/math]
- [math]\displaystyle{ 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 |
XOR basically means "exclusive or", meaning one input or the other must be true, but not both.
The XOR operator is written as [math]\displaystyle{ - }[/math] like this:
- [math]\displaystyle{ 0 - 0 = 0 }[/math]
- [math]\displaystyle{ 0 - 1 = 1 }[/math]
- [math]\displaystyle{ 1 - 0 = 1 }[/math]
- [math]\displaystyle{ 1 - 1 = 0 }[/math]
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:
- [math]\displaystyle{ \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]\displaystyle{ \overline{\mbox{A}} \cdot \overline{\mbox{B}} }[/math]
- [math]\displaystyle{ \mbox{A} + 1 = 1 }[/math]
- [math]\displaystyle{ \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]\displaystyle{ \overline{ANY} }[/math] |
| Any | 0 | [math]\displaystyle{ \overline{ANY} }[/math] | [math]\displaystyle{ \{Any\} }[/math] |
, if [math]\displaystyle{ ANY=\{x|\{x\}=\{\{TRUE\}\or\{\overline{TRUE}\}, \};\and (TRUE, 0) \vdash TRUE \and \overline{0} = \{x\} }[/math].Template:Citation needed
or if [math]\displaystyle{
ANY=\{x \|\{TRUE\}, \{\overline{TRUE}\} .\}, }[/math]=TRUE, TRUE.,
DeMorgan's laws
Augustus De Morgan found out that it is possible to change a [math]\displaystyle{ + }[/math] sign to a [math]\displaystyle{ \cdot }[/math] sign and make or break a bar. See the 2 examples below:
- [math]\displaystyle{ \overline{\mbox{A} + \mbox{B}} = \overline{\mbox {A}} \cdot \overline{\mbox{B}} }[/math]
- [math]\displaystyle{ \overline{\mbox{A} \cdot \mbox{B}} = \overline{\mbox {A}} + \overline{\mbox{B}} }[/math]
"Make/break the bar and change the sign."
Related pages
References
Other websites
- Boolean algebra on All About Circuits
- Boolean algebra Citizendium