Tool to calculate the dual of a Boolean logical expression. The dual being a complementary expression inverting addition and multiplication as well as 0 and 1.

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!

The dual of a Boolean or of a Boolean expression is obtained by applying 2 operations: replacing/interchanging the logical ORs by logical ANDs and vice versa and replacing/interchanging the logical 0s by logical 1s and vice versa.

Example: The dual of a+b is a.b and conversely the dual of a.b is a+b (duality principle)

It is possible that the value $ a $ itself has a dual, some note this dual $ a' $ (be careful not to confuse this notation with the boolean NOT unary operator)

How to note the dual of a boolean equation?

The dual of a boolean function $ F $ is sometimes denoted by $ Fˊ $ (not to be confused with the complement or NOT function) or $ F ^ d $.

Likewise 0 and 1 are dual, true and false are duals, ∧ and ∨ are dual.

What is the duality principle?

Every Boolean expression has a dual, the Boolean Duality principle means that every theorem or any computation has a dual equivalent.

By proving something in Boolean algebra, its dual is also proved.

Example:x+1=1 has for dual x.0=0

Source code

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

Need Help ?

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

Questions / Comments

Thanks to your feedback and relevant comments, dCode has developed the best 'Boolean Dual' tool, so feel free to write! Thank you!

Thanks to your feedback and relevant comments, dCode has developed the best 'Boolean Dual' tool, so feel free to write! Thank you!