When a negation surrounds a formula in Disjunctive Normal Form (DNF), is the formula still in DNF?
For example: Would $\lnot((\lnot X\land \lnot Y) \lor (\lnot X \land Y))$ be classed as being in DNF, even though it is surrounded by a negation?
Negation in Disjunctive Normal Form
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logic
boolean-algebra
disjunctive-normal-form
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0NO; see [DNF](https://en.wikipedia.org/wiki/Disjunctive_normal_form#Definition). – 2017-01-05
1 Answers
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The usual definition of a formula in DNF excludes this. It shall be a disjonction of conjonctions of litterals. However your form is very closed to CNF form according to Morgan's laws:
$$\neg((\neg X \land \neg Y) \lor (\neg X \land Y))\equiv(X \lor Y) \land ( X\lor \neg Y)$$
In general converting a formula in DNF to an equivalent form in CNF may lead to an exponential growth.