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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.

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Truth Table -

Tag(s) : Symbolic Computation, Electronics

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# Truth Table

## Find Equation from Truth Table

 Ordered 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 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:

ANOT A
01
10

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

### 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 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.

### How to find the equation from the 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:

ABX
000
011
101
110
here are the different calculations (which give the same result)

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

### 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 are the minterms?

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 $X = \sum{m(3)}$

### What are the maxterms?

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 $X = \sum{M(0,1,2)}$

## Source code

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