View source for Disjunctive normal form

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{{MSC|03B05}}

A propositional formula is said to be in ''disjunctive normal form'' if it is of the form
\begin{equation}\label{eq1}
  \bigvee_{i=1}^n \;\bigwedge_{j=1}^{m_i} C_{ij} ,
\end{equation}
where each $C_{ij}$ ($1,\ldots,n$; $j=1,\ldots,m_i$) is either a variable or the negation of a variable. The form \ref{eq1} is realizable (is a [[tautology]]) if and only if, for each $i$, $C_{i1},\ldots,C_{im_i}$ do not contain both the formulas $p$ and $\neg p$, where $p$ is any variable.  For any propositional formula $A$ it is possible to construct an equivalent disjunctive normal form $B$ containing the same variables as $A$. Such a formula $B$ is then said to be ''the disjunctive normal form'' of the formula $A$.

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The dual of a disjunctive normal form is a [[conjunctive normal form]]. Both are also used in the theory of Boolean functions (cf. [[Boolean functions, normal forms of|Boolean functions, normal forms of]]).