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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Truth Table
Truth Table Generator
Find Equation from Truth Table
Answers to Questions (FAQ)
What is a truth table? (Definition)
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.
How does the truth table calculator work?
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.
How to find the equation from the 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
What is the truth table for logical AND?
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 |
What is the truth table for logical OR?
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 |
What is the truth table for logical XOR?
What is the truth table for logical NAND?
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 |
What is the truth table for logical NOR?
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 |
What is the truth table for logical TRUE?
The truth table for the TRUE (=1) function is:
| A | TRUE |
|---|---|
| 0 | 1 |
| 1 | 1 |
What is the truth table for logical FALSE?
The truth table for the FALSE (=0) function is:
| A | FALSE |
|---|---|
| 0 | 0 |
| 1 | 0 |
What are the minterms?
What are the maxterms?
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)} $
Source code
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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 is the truth table for logical TRUE?
- What is the truth table for logical FALSE?
- What are the minterms?
- What are the maxterms?
