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i am trying to prove this statement, it is obvious, but i just cannot get the clue where and how to start to prove that this statement is true.

the statement is this: $(A \setminus C)\times(B \setminus D)\subset(A \times B)\setminus(C \times D)$

i am starting like this:

$x \in A \wedge x \notin C \times x \in B \wedge x \notin D $ ... but i dont know how to come to the idea that this is a subset of the right side of the first statement. :(

can someone give me a hint please?

thanks a lot

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up vote 6 down vote accepted

Pick an element in the left side, and show its in the right side.

Suppose $(x,y)\in (A\setminus C) \times (B\setminus D)$.

Then we have that $x\in A$, $x\notin C$ and $y\in B$, $y\notin D$.

So since $x\in A$ and $y\in B$ what can we say about $(x,y)$?

And since $x\notin C$ (and $y\notin D$) what can we say about $(x,y)$?

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+1 Very nicely done. – amWhy Nov 11 '12 at 18:48
wow, thanks, i got the $ clue $ :) – doniyor Nov 11 '12 at 18:51
@doniyor glad I could help! – Deven Ware Nov 11 '12 at 19:02

Actually, your starting point is false. You have to start with the definition of "Cartesian product ". So, $(x,y) \in (A-B)\times (B-D)$ must be your start point. Then, your next step will be: $x \in (A-B) \wedge y\in(B-D) $.

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