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Difference between revisions of "Disjunctive normal form"

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A propositional formula of the form
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033300/d0333001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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A canonical form for a [[propositional formula]].  A formula is said to be in ''disjunctive normal form'' if it is of the form
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\begin{equation}\label{eq1}
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  \bigvee_{i=1}^n \;\bigwedge_{j=1}^{m_i} C_{ij} ,
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\end{equation}
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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 $A$.
  
where each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033300/d0333002.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033300/d0333003.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033300/d0333004.png" />) is either a variable or the negation of a variable. The form (*) is realizable if and only if, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033300/d0333005.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033300/d0333006.png" /> do not contain both the formulas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033300/d0333007.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033300/d0333008.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033300/d0333009.png" /> is any variable. For any propositional formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033300/d03330010.png" /> it is possible to construct an equivalent disjunctive normal form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033300/d03330011.png" /> containing the same variables as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033300/d03330012.png" />. Such a formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033300/d03330013.png" /> is then said to be the disjunctive normal form of the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033300/d03330014.png" />.
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====Comments====
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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]]).
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The form \ref{eq1} may be referred to as a disjunctive form: for a given set of $m$ propositional variables $p_1,\ldots,p_m$, the normal form is that in which each term $\wedge C_{ij}$ contains exactly $m$ terms $C_{ij}$, each being either $p_j$ or $\neg p_j$, and in which no term is repeated. This form is then unique up to order. The formula may be read as expressing the rows of the [[truth table]] for a propositional formula, in which each term describes one particular row of the table, corresponding to an assignment of truth values to the $p_j$, and the disjunctive form corresponds to the truth value assignments for which the formula takes the value "true".
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====References====
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* Paul M. Cohn, ''Basic Algebra: Groups, Rings, and Fields'', Springer (2003) ISBN 1852335874

Latest revision as of 12:37, 29 November 2014

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

A canonical form for a propositional formula. A 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 $A$.

Comments

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).

The form \ref{eq1} may be referred to as a disjunctive form: for a given set of $m$ propositional variables $p_1,\ldots,p_m$, the normal form is that in which each term $\wedge C_{ij}$ contains exactly $m$ terms $C_{ij}$, each being either $p_j$ or $\neg p_j$, and in which no term is repeated. This form is then unique up to order. The formula may be read as expressing the rows of the truth table for a propositional formula, in which each term describes one particular row of the table, corresponding to an assignment of truth values to the $p_j$, and the disjunctive form corresponds to the truth value assignments for which the formula takes the value "true".

References

  • Paul M. Cohn, Basic Algebra: Groups, Rings, and Fields, Springer (2003) ISBN 1852335874
How to Cite This Entry:
Disjunctive normal form. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Disjunctive_normal_form&oldid=14566
This article was adapted from an original article by S.K. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article