Tool to generate logical truth tables. In Boolean algebra or electronics, logical truth tables allow to define 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 |

The 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)} $

The 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' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any 'Truth Table' algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any 'Truth Table' function (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and no data download, script, copy-paste, or API access for 'Truth Table' will be for free, same for offline use on PC, tablet, iPhone or Android ! dCode is free and online.

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- Truth Table Generator
- Find Equation from Truth Table
- What is a truth table?
- 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 minterms?
- What are maxterms?

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