# DeMorgans Laws formulation

1. Aug 8, 2014

### 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
$\, \,\begin{Bmatrix} A-(B\cup C)\, \end{Bmatrix}\, and\, \begin{Bmatrix} (A-B)\cup (A-C) \end{Bmatrix}\, \, \,$

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

$\therefore \, \, \forall x\, \in\begin{Bmatrix} A-(B\cup C)\, \end{Bmatrix}, x\in A$

Now,

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

$\Rightarrow \forall x\in \begin{Bmatrix} (A-B)\cap (A-C), \, x \in A \end{Bmatrix}$$x \in A$

Thus: $\begin{Bmatrix} A-(B \cup C) \end{Bmatrix} =\begin{Bmatrix} (A-B)\cap (A-C) \end{Bmatrix}$

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

Thanks again!

Last edited: Aug 8, 2014
2. Aug 8, 2014

### HallsofIvy

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

3. Aug 8, 2014

### 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:

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

$\therefore \, \, \forall x\, \in\begin{Bmatrix} A-(B\cup C)\, \end{Bmatrix}, x\in A$

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

Last edited: Aug 8, 2014