What is the correct form of De Morgan's Law in logic?

Question

According to wikipedia (link), Morgan's Law is:

$$¬ (P \wedge Q) \Rightarrow (¬P) \vee (¬Q)$$

But if you scroll down to 8.2.2 on this page (link), it says that Morgan's Law works as follow:

$$¬ (P \wedge Q) \Rightarrow ¬P \vee Q$$

I believe wikipedia but I have had some excercices where they use Morgan's Law like the other website. Which method is correct?

Answer 1

DeMorgan's Law is really DeMorgan's Laws:

$$\lnot(P \land Q) \equiv \lnot P \lor \lnot Q\tag{1}$$

$$\lnot (P \lor Q) \equiv \lnot P \land \lnot Q\tag{2}$$

Note that in both $(1)$ and $(2)$, the left-hand side and right-hand side are logically equivalent, meaning each side of the equivalence implies the other.

So, the "other website" is most certainly incorrect. It's always good to confirm or challenge what you read on the internet!

Answer 2

Good catch, it is definitely not true that $$\neg(P \wedge Q) \equiv \neg P \vee Q$$ The website you linked seems pretty unprofessional and I'm guessing whoever wrote it made a (critical) typo. Hopefully nobody else out there saw that and believes it is the correct statement of De Morgan's Law.

Answer 3

A visual representation:
Basically, collecting the or flips the or into an and.
In the second row, collecting the or transforms Q into not Q.
If the wrong version reported in the website reminds you of some exercises, maybe something similar to the second row was happening! In both rows, what's true is the wikipedia version (expressed in the first row)