Search for a tool
Truth Table

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.

Results

Truth Table -

Tag(s) : Symbolic Computation, Electronics

Share dCode and more

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Feedback and suggestions are welcome so that dCode offers the best 'Truth Table' tool for free! Thank you!

# Truth Table

## Find Equation from Truth Table

Indicate only the output values of the function (the last column from the boolean truth table)

 Table ordered (input values) From 0,…,0 to 1,…,1 From 1,…,1 to 0,…,0
 Boolean Notation Literal (AND, OR, NOT) Logical (∧, ∨, ¬) Programming (&&, ||, ~) Algebraic (*, +, !)

### 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):

ANOT A
01
10

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
000
011
101
110

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:

ABA AND B
000
010
100
111

### What is the truth table for logical OR?

The truth table for the OR function is:

ABA OR B
000
011
101
111

### What is the truth table for logical XOR?

The truth table for the XOR function is:

ABA XOR B
000
011
101
110

### What is the truth table for logical NAND?

The truth table for the NAND function is:

ABA NAND B
001
011
101
110

### What is the truth table for logical NOR?

The truth table for the NOR function is:

ABA NOR B
001
010
100
110

### What is the truth table for logical TRUE?

The truth table for the TRUE (=1) function is:

ATRUE
01
11

### What is the truth table for logical FALSE?

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

AFALSE
00
10

### What are the minterms?

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

Example: $X = a \oplus b$ (XOR) the truth table has 2 TRUE outputs at 2nd row (indexed 1) and 3rd row (indexed 2), so $X = \sum{m(1,2)}$

### 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

dCode retains ownership of the "Truth Table" source code. Except explicit open source licence (indicated Creative Commons / free), the "Truth Table" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, 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, or API access for "Truth Table" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.

## Cite dCode

The copy-paste of the page "Truth Table" or any of its results, is allowed (even for commercial purposes) as long as you cite dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Truth Table on dCode.fr [online website], retrieved on 2023-12-06, https://www.dcode.fr/boolean-truth-table

## Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!