Formulating and Proving DeMorgan's Laws in Set Theory

[FeynmanIsCool]

Hello,
I am working through Munkres Topology (not for a class). It asks the reader to formulate and proove DeMorgans Laws. I am new to proofs, so I was wondering if this is what the book is asking. Any help would be appreciated!

assume two sets
\, \,\begin{Bmatrix}<br /> A-(B\cup C)\,<br /> \end{Bmatrix}\, and\, \begin{Bmatrix}<br /> (A-B)\cup (A-C)<br /> \end{Bmatrix}\, \, \,

\forall x\in (B\cup C), x\in B\, or\, x\in C \, or \, both

\therefore \, \, \forall x\, \in\begin{Bmatrix}<br /> A-(B\cup C)\,<br /> \end{Bmatrix}, x\in A

Now,

\forall x\in (A-B), \, x\in A\, \, and\, \, \forall x\in (A-C), \, \, x\in A

\Rightarrow \forall x\in \begin{Bmatrix}<br /> (A-B)\cap (A-C), \, x \in A<br /> \end{Bmatrix}x \in AThus: \begin{Bmatrix}<br /> A-(B \cup C)<br /> \end{Bmatrix}<br /> =\begin{Bmatrix}<br /> (A-B)\cap (A-C)<br /> \end{Bmatrix}

Does this proove DeMorgans Law (just the first one)? Formally?

Thanks again!

 

[HallsofIvy]

If x\in"A- anything" then x\in A. You haven't said anything about x NOT being in the other sets.

 

[FeynmanIsCool]

so I also need to state: x\notin B and x\notin C ?
So if I tagged those statements onto the proof after I state x\in A, then its good?

Im just a little confused, by saying:

\forall x\in (B\cup C), x\in B\, or\, x\in C \, or \, both

\therefore \, \, \forall x\, \in\begin{Bmatrix}<br /> A-(B\cup C)\,<br /> \end{Bmatrix}, x\in A

Aren't I stating that I am taking out all elements of B and C, thus all elements left are elements of A?