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Let $A$ and $B$ be subsets of $X$ and $Y$, respectively. I've seen this written in a proof: $$A \times B = X\times Y - ((X-A)\times B \cup A\times(Y-B))$$

I think this is not correct, and I want to confirm it. I think one must have $$A \times B = X\times Y - ((X-A)\times B \cup A\times(Y-B)\cup ((X-A)\times(Y-B)))$$

In fact, in the first case one would have that $(x,y)\in A\times B$ for $x\in (X-A)$ and $y \in (Y-B)$, which is incorrect.

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    After seeing the proof again, I noticed that it was written $$A \times B = X\times Y - ((X-A)\times Y \cup X\times(Y-B))$$ after all. This excludes the case I was worried about. I leave the question here anyway because I would still like someone to confirm it. – 2017-01-16
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    Is $A - B$ the set-theoretic difference of $A$ and $B$ (all elements of $A$ that are not in $B$)? – 2017-01-16
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    Yes, it is the set-theoretic difference. – 2017-01-16

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The formula from your comment is correct.

Observe that for sets $C, D, E, F$ we have $$ C-(D\cup E)=(C-D)\cap (C-E),$$ $$(C-D)\times E=C\times E - D\times E,$$ and $$(C\times D)\cap (E\times F)=(C\cap E)\times (D\cap F).$$

Using these three facts you obtain $$X\times Y-((X-A)\times Y\cup X\times (Y-B))=(X\times Y-((X-A)\times Y))\cap (X\times Y-(X\times (Y-B)))= ((X-(X-A))\times Y)\cap (X\times (Y-(Y-B)))=(A\times Y)\cap(X\times B)=A\times B.$$

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    I would like to practice calculations with set operations (differences, cartesian products, unions and intersections). Could you tell me a good place to look for some exercises like those with solutions? Everything I find focuses on the theory (of sets), not on the calculations. – 2017-01-17
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    Pfff, I'm afraid that not:( I myself learned that in school, but not from a book. – 2017-01-17