DeMorgan's rule is yet another rule uv inference used in propositional logic. It's a rather complicated one, so read it carefully.

~(P*Q) = (~P^~Q)
And
~(P^Q) = (~P*~Q)

Check out Everything Logic Symbols if you can't understand all this.

In everyday language, these read as:

"It is not the case that I have an apple and an orange" is the same as "I don't have an apple, or I don't have an orange"

And...

"It is not the case that I have an apple or and orange" is the same as "I don't have an apple, and I don't have an orange".

DeMorgan's rule is often abbreviated DeM when doing logic problems. It is also known as DeMorgan's Law or DeMorgan's theorem. It is named after the 19th century logician Augustus De Morgan, who was the first to state it formally.

Other rules of inference include Modus Tollens, Modus Ponens, and commutativity.

A handy way of remembering this rule is thus: "Break the line, change the sign"

You'll have to use your imagination on this. If I have ¬(A∧B), imagine that that is AB with a line over it. That is equivalent to ¬A∨¬B, which would be A with an overline and B with a separate overline. When the line is broken, the sign is switched. So if you had ¬(¬C∨D), the two overlines attached to C would counteract each other, breaking the line, and the result would be C∧¬D.

References: my Digital Systems teacher