Tool to generate logical truth tables. In Boolean algebra or electronics, logical truth tables allow defining a function / gate / element / component according to its inputs and outputs.

Truth Table - dCode

Tag(s) : Symbolic Computation, Electronics

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A **truth table** is a table representing the output boolean values of a logical expression based on their entries. The table thus presents all the possible combinations of the input logical variables (generally 0 / FALSE and 1 / TRUE) and the result of the equation as output.

__Example:__ The table of the logical NOT function:

A | NOT A |
---|---|

0 | 1 |

1 | 0 |

Every electronic circuit is associated with a **truth table** which describes it.

dCode **truth table** generator interprets the Boolean logical expression and calculates, using Boolean algebra, all the possible combinations of 0 and 1 for each variable (among the Boolean variables requested) in order to convert the boolean expression and make the **truth table**.

dCode also makes it possible to find the Boolean logic function/expression from a **truth table**.

There are 2 methods to find the Boolean equation from the **truth table**, either by starting from values `0` (calculation of Maxterms) or by starting from values `1` (calculation of Minterms).

__Example:__ The **truth table** is:

A | B | X |
---|---|---|

0 | 0 | 0 |

0 | 1 | 1 |

1 | 0 | 1 |

1 | 1 | 0 |

Calculation from the values `1` of the **truth table** (Minterms): for each `1`, write on line the values of the corresponding entries separated by logical `AND`, then group these lines with a logical `OR`.

__Example:__ Lines 2 and 3 are equal to `1`, line 2 is written as `A AND NOT(B)`, line 3 is written as `NOT(A) AND B` and therefore the equation is `(A AND NOT(B)) OR (NOT(A) AND B)` which possibly simplifies to `A XOR B`

Calculation from the values `0` of the **truth table** (Maxterms): for each `0`, write on line the values of the corresponding inputs separated by logical `OR`, then each line separated by a logical `AND`.

__Example:__ Lines 1 and 4 are equal to `0`, line 1 is written as `A OR B`, line 4 is written as `NOT(A) OR NOT(B)` and therefore the equation is `(A OR B) AND (NOT(A) OR NOT(B))` which possibly simplifies to `A XOR B`

The **truth table** for the AND function is:

A | B | A AND B |
---|---|---|

0 | 0 | 0 |

0 | 1 | 0 |

1 | 0 | 0 |

1 | 1 | 1 |

The **truth table** for the OR function is:

A | B | A OR B |
---|---|---|

0 | 0 | 0 |

0 | 1 | 1 |

1 | 0 | 1 |

1 | 1 | 1 |

The **truth table** for the NAND function is:

A | B | A NAND B |
---|---|---|

0 | 0 | 1 |

0 | 1 | 1 |

1 | 0 | 1 |

1 | 1 | 0 |

The **truth table** for the NOR function is:

A | B | A NOR B |
---|---|---|

0 | 0 | 1 |

0 | 1 | 0 |

1 | 0 | 0 |

1 | 1 | 0 |

Minterms $ m $ are the row numbers of the table which have a logical 1 output (row numbering from 0).

__Example:__ $ X = a + b $ the **truth table** has 1 output TRUE on the 3rd row, so $ X = \sum{m(3)} $

Maxterms $ M $ are the row numbers of the table which have a logical 0 output (row numbering from 0).

__Example:__ $ X = a + b $ the **truth table** has 3 output FALSE in the 3 first rows noted 0, 1 et 2 so $ X = \sum{M(0,1,2)} $

dCode retains ownership of the online "Truth Table" source code. Except explicit open source licence (indicated CC / Creative Commons / free), the "Truth Table" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or the "Truth Table" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, copy-paste, or API access for "Truth Table" are not public, same for offline use on PC, tablet, iPhone or Android ! Remainder : dCode is free to use.

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- Truth Table Generator
- Find Equation from Truth Table
- What is a truth table? (Definition)
- How does the truth table calculator work?
- How to find the equation from the truth table?
- What is the truth table for logical AND?
- What is the truth table for logical OR?
- What is the truth table for logical XOR?
- What is the truth table for logical NAND?
- What is the truth table for logical NOR?
- What are the minterms?
- What are the maxterms?

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Source : https://www.dcode.fr/boolean-truth-table

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