Boolean Expressions Calculator

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 ! for logical NOT.

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) are not taken into account, please write a XOR b

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

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

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:

$$ \overline{(A \land B)} \leftrightarrow (\overline{A})\lor (\overline{B}) \iff \bar{AB} = \bar{A} + \bar{B} $$

$$ \overline{(A \lor B)} \leftrightarrow (\overline{A}) \land (\overline{B}) \iff \bar{A+B} = \bar{A} . \bar{B} $$

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)

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.