Search for a tool
Boolean Expressions Calculator

Tool/Calculator to simplify or minify Boolean expressions (Boolean algebra) containing logical expressions with AND, OR, NOT, XOR.

Results

Boolean Expressions Calculator -

Tag(s) : Symbolic Computation, Electronics

Share
Share
dCode and you

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 community Discord for help requests!


Thanks to your feedback and relevant comments, dCode has developped the best Boolean Expressions Calculator tool, so feel free to write! Thank you !

Boolean Expressions Calculator

Boolean Expressions Simplifier










Tool/Calculator to simplify or minify Boolean expressions (Boolean algebra) containing logical expressions with AND, OR, NOT, XOR.

Answers to Questions

How to simplify / minify a boolean expression?

The simplification of Boolean Equations can use different methods: besides the classical development via associativity, commutativity, distributivity, etc., Truth tables or Venn diagrams provide a good overview of the expressions.

dCode allows several syntaxes:

Algebraic notation

!(ab(c+!d))+!b with implicit multiplication ab = a AND b and ! (exclamation) for logical NOT.

Logic/Computer notation

Example: !(a&&b&&(c||!d))||!b with double character & (ampersand) for AND and the double character | (pipe, vertical bar) for logical OR.

Literal notation

Example: NOT (a AND b AND (c OR NOT d)) OR NOT b

There may be several minimal representations for the same expression, dCode provides a solution and output an algebraic notation.

Functional notation XOR(a,b) is not handled, please write a XOR b

What are boolean algebra simplifications methods?

Boolean algebra has many properties (boolen laws):

1 - Identity element : $ 0 $ is neutral for logical OR while $ 1 $ is neutral for logical AND

$$ a + 0 = a \\ a.1 = a $$

2 - Absorption : $ 1 $ is absorbing for logical OR while $ 0 $ is absorbing for logical AND

$$ a + 1 = 1 \\ a.0 = 0 $$

3 - Idempotence : applying multiple times the same operation does not change the value

$$ a + a = a + a + \cdots + a = a \\ a . a = a . a . \cdots . a = a $$

4 - Involution or double complement : the opposite of the opposite of $ a $ est $ a $

$$ a = \overline{\overline{a}} = !(!a) $$

5 - Complementarity by Contradiction : $ a $ AND $ \text{not}(a) $ is impossible, so is false and is $ 0 $

$$ a . \overline{a} = 0 $$

6 - Complementarity by excluded third : $ a $ OR $ \text{not}(a) $ is always true, so is $ 1 $

$$ a + \overline{a} = 1 $$

7 - Associativity law : parenthesis are useless between same operators

$$ a.(b.c) = (a.b).c = a.b.c \\ a+(b+c) = (a+b)+c = a+b+c $$

8 - Commutativity law : the order does not matter

$$ a.b = b.a \\ a+b = b+a $$

9 - Distributivity law : AND is distributed over OR but also OR is distributed over AND

$$ a.(b+c) = a.b + a.c \\ a+(b.c) = (a+b).(a+c) $$

10 - De Morgan laws (see below for more details)

$$ \overline{a+b} = \overline{a}.\overline{b} \\ \overline{a.b} = \overline{a}+\overline{b} $$

11 - Other simplifications by combinations of the above ones

$$ a.(a+b) = a \\ a+(a.b) = a \\ (a.b) + (a.!b) = a \\ (a+b).(a+!b) = a \\ a + (!a.b) = a + b \\ a.(!a + b) = a.b \\ $ a.b + \overline{a}.c = a.b + \overline{a}.c + b.c $$

How to show/demonstrate that 2 boolean expression are equal?

Method 1: simplify them until you get the same writing in boolean algebra.

Method 2: by calculating their truth table which should be identical.

What is De Morgan's law?

De Morgan's laws are often used to rewrite logical expressions. They are generally stated: not (a and b) = (not a) or (not b) and not (a or b) = (not a) and (not b). Here are the equivalent logical entries:

$$ \overline{(a \land b)} \leftrightarrow (\overline{a})\lor (\overline{b}) \iff \bar{AB} = \bar{a} + \bar{b} $$

$$ \overline{(a \lor b)} \leftrightarrow (\overline{a}) \land (\overline{b}) \iff \bar{a+b} = \bar{a} . \bar{b} $$

What is Disjunctive or Conjunctive Normal Form?

In logic, it is possible to use different formats to ensure better readability or usability.

The normal disjunctive form (DNF) uses a sum of products:

Example: (a&&c)||b

The normal conjunctive form (CNF) or clausal form uses a product of sums:

Example: (a||b)&&(b||c)

How to show step by step calculations?

The calculation steps, such as a human imagines them, do not exist for the solver. The operations performed are binary bit-by-bit and do not correspond to those performed during a resolution with a pencil and paper.

Source code

dCode retains ownership of the online 'Boolean Expressions Calculator' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) no data, script or API access will be for free, same for Boolean Expressions Calculator download for offline use on PC, tablet, iPhone or Android !

Need Help ?

Please, check our community Discord for help requests!

Questions / Comments

Thanks to your feedback and relevant comments, dCode has developped the best Boolean Expressions Calculator tool, so feel free to write! Thank you !


Source : https://www.dcode.fr/boolean-expressions-calculator
© 2020 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.
Feedback