Disjunctive normal form

2020 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