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 displays all the possible combinations of the $ n $ input logical variables (generally 0 / FALSE and 1 / TRUE over $ n $ columns) and the result of the equation as output (last column).

__Example:__ The table of the logical NOT function with 1 input `A` (1st column) and the output value `NOT A` (last column):

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

0 | 1 |

1 | 0 |

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

Usually the array starts with input values at $ 0 $ and ends with input values at $ 1 $ but sometimes it's reversed.

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 make the truth table.

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

The only important column of the truth table is the last one, which describes the output values (the first columns are always identical for a given number of inputs) and which allows to convert into the Boolean expression.

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

__Example:__ The output values are `0,1,1,0`, (and the table is ordered from 00 to 11), so the truth table is:

input A | input B | output X |
---|---|---|

0 | 0 | 0 |

0 | 1 | 1 |

1 | 0 | 1 |

1 | 1 | 0 |

Here are the different calculations (which give the same result for the given example)

Calculation from the output values `1` of the truth table (Minterms): for each `1`, write the corresponding input values separated by logical `AND`, then group these results with a logical `OR`.

__Example:__ Rows 2 and 3 are equal to `1`, row 2 is written as `A AND NOT(B)` (because `A` is `1` and `B` is `0`), row 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 output values `0` of the truth table (Maxterms): for each `0`, write the corresponding input values separated by logical `OR`, then group these results with a logical `AND`.

__Example:__ Rows 1 and 4 are equal to `0`, row 1 is written as `A OR B`, row 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 truth table for the TRUE (=1) function is:

A | TRUE |
---|---|

0 | 1 |

1 | 1 |

The truth table for the FALSE (=0) function is:

A | FALSE |
---|---|

0 | 0 |

1 | 0 |

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

__Example:__ $ X = a \oplus b $ (XOR) the truth table has 2 outputs FALSE at the first and last lines denoted 0, and 3 therefore $ X = \prod{M(0,3)} $

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Cite as source (bibliography):

*Truth Table* on dCode.fr [online website], retrieved on 2023-12-06,

- 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 is the truth table for logical TRUE?
- What is the truth table for logical FALSE?
- What are the minterms?
- What are the maxterms?

truth,table,boolean,logical,electronic,boole

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