DeMorgan's Rule

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

"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".

(These also work the other way around.)

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

See also: DeMorgan's rule, Modus Ponens, 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