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
[itex]\, \,\begin{Bmatrix}
A-(B\cup C)\,
\end{Bmatrix}\, and\, \begin{Bmatrix}
(A-B)\cup (A-C)
\end{Bmatrix}\, \, \, [/itex]

[itex]\forall x\in (B\cup C), x\in B\, or\, x\in C \, or \, both[/itex]

[itex]\therefore \, \, \forall x\, \in\begin{Bmatrix}
A-(B\cup C)\,
\end{Bmatrix}, x\in A[/itex]

Now,

[itex]\forall x\in (A-B), \, x\in A\, \, and\, \, \forall x\in (A-C), \, \, x\in A[/itex]

[itex]\Rightarrow \forall x\in \begin{Bmatrix}
(A-B)\cap (A-C), \, x \in A
\end{Bmatrix}[/itex][itex] x \in A[/itex]Thus: [itex]\begin{Bmatrix}
A-(B \cup C)
\end{Bmatrix}
=\begin{Bmatrix}
(A-B)\cap (A-C)
\end{Bmatrix}[/itex]

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

Thanks again!

 

[HallsofIvy]

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

 

[FeynmanIsCool]

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

Im just a little confused, by saying:

[itex]\forall x\in (B\cup C), x\in B\, or\, x\in C \, or \, both[/itex]

[itex]\therefore \, \, \forall x\, \in\begin{Bmatrix}
A-(B\cup C)\,
\end{Bmatrix}, x\in A[/itex]

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