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# Conjunctive Normal Form

A statement is in conjunctive normal form if it is a conjunction (sequence of ANDs) consisting of one or more conjuncts, each of which is a disjunction (OR) of one or more literals (i.e., statement letters and negations of statement letters; Mendelson 1997, p. 30). Examples of conjunctive normal forms include

 (1) (2) (3) (4)

where denotes OR, denotes AND, and denotes NOT (Mendelson 1997, p. 30).

Every statement in logic consisting of a combination of multiple , , and s can be written in conjunctive normal form.

An expression can be put in conjunctive normal form using the Wolfram Language using the following code:

```  ConjunctiveNormalForm[f_] :=
Not[LogicalExpand[Not[f]]] //. {
Not[a_Or] :> And @@ (Not /@ List @@ a),
Not[a_And] :> Or @@ (Not /@ List @@ a)
}```

AND, Disjunctive Normal Form, Literal, Negation, Normal Form, OR, Statement Letter

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## References

Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, p. 30, 1997.

## Referenced on Wolfram|Alpha

Conjunctive Normal Form

## Cite this as:

Weisstein, Eric W. "Conjunctive Normal Form." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ConjunctiveNormalForm.html