Truth Table

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:

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:

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:

The truth table for the OR function is:

The truth table for the XOR function is:

The truth table for the NAND function is:

The truth table for the NOR function is:

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)} $