Product of sets

[borkborkmath]

"Let \(\displaystyle X \subset A\), \(\displaystyle Y \subset B\). Prove that:
\(\displaystyle C(X \times Y) = A \times C(Y) \cup C(X) \times B\)"


Where C() is the complement and x is the cartesian product.

 

[pka]

"Let \(\displaystyle X \subset A\), \(\displaystyle Y \subset B\). Prove that:
\(\displaystyle C(X \times Y) = A \times C(Y) \cup C(X) \times B\)"

Here is an outline.
If \(\displaystyle (p,q)\in C(X\times Y)\) then \(\displaystyle p\notin X\text{ or }q\notin Y\).
So \(\displaystyle p\in A\cap C(X)\text{ or }q\in C(Y)\cap B.\)

That means \(\displaystyle (p,q)\in\left[(A\times C(Y))\cup(C(X)\times B)\right]\)

 

[borkborkmath]

Thanks for your help

In your second line of work, it is the intersection because X\(\displaystyle \subset\)A and Y\(\displaystyle \subset\)B, right? Because if its in the C() it must still be in the whole set.

Then to show containment in the other way...

let (x,y) \(\displaystyle \in\) AxC(Y)\(\displaystyle \cup\)C(X)xB

does this imply that x\(\displaystyle \in\)A or x\(\displaystyle \in\)C(X) and y\(\displaystyle \in\)C(Y) or y\(\displaystyle \in\)B

and if so, how do I get from there to (x,y)\(\displaystyle \in\)C(XxY)?

 

[pka]

In your second line of work, it is the intersection because X\(\displaystyle \subset\)A and Y\(\displaystyle \subset\)B, right? Because if its in the C() it must still be in the whole set.

I have no idea you could mean by that.
Those are really relative complements: \(\displaystyle C(X)=A\setminus X\)

Then to show containment in the other way...let (x,y) \(\displaystyle \in\) AxC(Y)\(\displaystyle \cup\)C(X)xB

To say \(\displaystyle (x,y)\in[(A\times C(Y))\cup (C(X)\times Y)]\)
means \(\displaystyle (x,y)\in(A\times C(Y))\text{ or }(x,y)\in (C(X)\times Y)\)

BTW: you can clean up your LaTeX.
[TEX](x,y)\in[(A\times C(Y))\cup (C(X)\times Y)][/TEX] gives
\(\displaystyle (x,y)\in[(A\times C(Y))\cup (C(X)\times Y)]\)