# Disjunctive normal form

2010 Mathematics Subject Classification: *Primary:* 03B05 [MSN][ZBL]

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 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 $A$.

**How to Cite This Entry:**

Disjunctive normal form.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Disjunctive_normal_form&oldid=27312