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De Morgan's second law makes sense: not p and q is the same as not p or not q. However, when I tried to construct a sentence as an example of the law, it seemed not true. For example:

Sequa does not like pie and cake. Sequa does not like pie or does not like cake.

These sentences appear to not be the same, for the first sentences states that she does not like pie and cake. The second states that she does not like one or the other, not necessarily that she does not like both.

Am I giving a incorrect example?

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up vote 12 down vote accepted

The "not" of the first sentence seems to be asserting that "Sequa $\lnot$(likes pie $\land$ likes cake)."

That means she does not like the combination (pie and cake), or perhaps more clearly, that "it is not the case that Sequa (likes both pie and cake). This is equivalent to the statement: "Sequa does not like pie or Sequa does not like cake." (She must dislike at least one of the two).

Then you have a correct application of DeMorgan's second law: $\quad\lnot(p \land q) \equiv \lnot p \lor \lnot q.$

If you want to say "Sequa does not like cake and she does not like pie", then you would want to state "Sequa does not(like pie or like cake). That is, she doesn't like either one:

This is an application of DeMorgan's first law: $\quad \lnot(p \lor q)\equiv \lnot p \land \lnot q.$

English is ambiguous and so logic doesn't always capture well the different manners in which "not", "and", "or", and other connectives are actually used in natural language.

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Very nice + 1 for a new badge! :-) – Amzoti May 2 '13 at 0:31

De Morgan's law states that $\neg (p\wedge q)$ if and only if $\neg p \vee \neg q$.

Let $p$ be the statement that Sequa likes pie, and $q$ the statement that Sequa likes cake. Then we can rewrite $\neg (p\wedge q)$ as "It is not true that Sequa likes both pie and cake" and $\neg p \vee \neg q$ as "Sequa does not like pie or does not like cake."

The second sentence is as you wrote it, but the first is a bit ambiguous in Enlgish. Does "Sequa does not like pie and cake" mean that she dislikes both, or just that it is not true that she likes both (and thus could still like one if she dislikes the other)? So your example is not wrong, per se, but you should be careful to make sure that your statement in English clearly matches the intended logical meaning.

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so $\neg (p\wedge q)$ means that if you show Sequa pie and cake, and give her only one chance to say like or dislike (and she says dislike), it's equivalent to her saying she either doesn't like one of them or both. – CodyBugstein Dec 9 '12 at 4:16
Yes, it means it is not true that she likes both, so she must dislike at least one. – Brett Frankel Dec 9 '12 at 4:24

(Revised) Your example is correct. Perhaps you are thinking that the second sentence should be: Sequa does not like pie, AND does not like cake? That would be the case if your first sentence had been: Sequa does not like pie OR cake.

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