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Let $E= A_1 \times A_2 \times \dots A_d$ and $F = B_1 \times B_2 \times \dots B_d$. Is there a way to express the set $E \setminus F$ as an union somehow?

For say $d=2$ case one has $(A_1 \times A_2) \setminus (B_1 \times B_2) = (A_1 \times (A_2 \setminus B_2)) \cup ((A_1 \setminus B_1) \times B_2)$, but I have trouble expressing this with more elements. What should I consider here?

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How about:$$A_1\times\cdots\times A_i\times\cdots\times A_d-B_1\times\cdots\times B_i\times\cdots\times B_d=\bigcup_{i=1}^dA_1\times\cdots\times (A_i\setminus B_i)\times\cdots\times A_d$$

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