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I want to write the following in a formula. I have a set of things A. Each of the things has its own set of things B_A, which are characterized by an area (m²). There is another variable, let's call it maximum allowed content k > 0. For each element of A I want to find the k-largest elements of B_A and then add those largest elements over A.

How do I write that?

Can I write the condition next to the formula as in the Example Formula

Sebastian E
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    If the $B$ are finite, let each $B \in A$ be indexed by positive integers, satisfying $i<j \implies |b_i|\ge|b_j|$. Your sum is$$\sum_{B\in A} \sum_{\substack{b_i \in B \ 1\le i \le k}} b_i$$ – Joe Nov 26 '21 at 12:32
  • @Joe, thank you. Can I just write the condition (i<j --> ...) next to the formula? I am a bit confused about the j since it seems to not be defined anywhere. Shouldn't it be defined more precisely, like 1<=j<=k? – Sebastian E Nov 26 '21 at 12:37
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    The $i$ and $j$ are dummy variables standing for any indices. More formally the condition would be written $\forall i,,j ,i<j \implies |b_i|\ge|b_j|$, and $i$ and $j$ are called bound variables. The condition explains the notation for the summation. It needs to be before the summation, but not combined with it into one statement. – Joe Nov 26 '21 at 12:54
  • Again, thank you. Is there a sign I can put between "\forall i,j" and "i<j", e.g., a comma. What I ment with "next to it" was, what is often shown in textbooks, is that wrong? I have attached an attachment to my question with a picture of what I ment. – Sebastian E Nov 26 '21 at 15:29
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    I'm not sure if there is a convention, or what the notation options are. Perhaps $$\forall i,j \in {1,\ldots,|B|} , (i<j)\implies |b_i|\ge|b_j|$$ – Joe Nov 26 '21 at 18:52
  • See this post for example Different standards for writing down logical quantifiers in a formal way – Joe Nov 26 '21 at 18:58
  • @Joe Thank you. Is there a way to say "if" with maths-symbols? I feel like this is missing between ∀i,j∈{1,…,|B|} and (i<j)⟹|bi|≥|bj|. – Sebastian E Dec 07 '21 at 11:28
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    The implication arrow, $A \implies B$, which can be read "A implies B", can also be read as "If A, then B." In the above formula, it would be read "For all $i$ and $j$ from 1 to the cardinality of B, if $i$ is less than $j$, then the cardinality of $b_i$ is greater than or equal to the cardinality of $b_j$." – Joe Dec 07 '21 at 12:20
  • I understand your point and it is of course true. but is there maybe a sign for "if"? – Sebastian E Dec 08 '21 at 14:40
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    I don't think there is a mathematical symbol for "if", other than the implication arrow. See the posts https://math.stackexchange.com/questions/3905499/is-there-a-math-symbol-meaning-if-or-when *and* https://math.stackexchange.com/questions/1004734/symbol-for-when – Joe Dec 08 '21 at 16:28

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