Is x not in A or B AND not in C?

[kernelpenguin]

I'm trying to prove something small with set theory and since I'm new to it, I've run into a problem. I can't understand what the following means exactly and how to proceed further. Or where the mistake is, if there is one. I think there is, because it seems... freaky.

$$x\notin\left(\left(\left(A\cup B\right)\setminus\left(A\cap B\right)\right)\cap C\right)$$
$$x\notin\left(\left(A\cup B\right)\setminus\left(A\cap B\right)\right)\wedge x\notin C$$
$$\left(x\notin\left(A\cup B\right)\wedge x\in\left(A\cap B\right)\right)\wedge x\notin C$$
$$\left(\left(\left(x\notin A\right)\vee\left(x\notin B\right)\right)\wedge\left(\left(x\in A\right)\wedge\left(x\in B\right)\right)\right)\wedge x\notin C$$

I'd post the entire thing of which this is a small part of, but that's my homework and I don't want to get into the habit of having other people do my homework for me. Plus I want to learn how and why it works, not just do it.

 

[matt grime]

consider this

x is not an element of UnV
x is not an element of U AND x is not an element of V

you've said those two statements are equivalent (I think, since you've not actually said what your deductions are from line to line). find a counter example to show this is false.

negation switches conjunction and disjunction, or union and intersection.

similar observations hold for the other steps in your reasoning.