Let $X, Y_{1}, ..., Y_{p}$ be nonempty subsets of $R^n, n \geq 1$. Can I express the set $$ \left\{x + \sum_{j = 1}^{p}v_j \mid x \in X, v_j \in Y_j, (v_1)_{n} = \dots = (v_p)_{n}\right\} $$ as a sum of sets, i.e $A + B = \{a + b \mid a \in A, b \in B\}$?
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2I think you're missing $X$ in the definition. And some triviality clause, otherwise $B={{\vec 0}}$ would do the trick. – Nikolaj-K May 22 '23 at 17:06
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$v_{1_n}$? I don't know what you meant be that. – JonathanZ May 22 '23 at 17:13
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@JonathanZsupportsMonicaC It's the n-th coordinate of the vector $v_1$ – Crostul May 22 '23 at 17:14
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Ah, $(v_1)n$, not $v{{1_n}}$. – JonathanZ May 22 '23 at 17:16
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Thanks for the remarks. I've rectified the definition. As for trivialities, I require A = X and B to be itself a sum of sets involving the $Y_j$'s. – AMfrn May 22 '23 at 17:40