# Notation for existential qualifier without repeats

Apologies for any incorrect terminology, my background is in CS, not math. I'm trying to figure out how to notate that in a list of sets (of numbers), the sets can be paired such that in each pair of sets, the sum of the members of each set are equal. What I currently have is: $$\forall_A \exists_B (\sum_{x \in A} x = \sum_{y \in B} y)$$ where A and B are sets in the list. The problem (as I see it) is that this would be true for a list of three sets which all have equal sums, while what I am trying to write is that once a set is used in one pair, it cannot be in another (the list would have to include an even number of sets), and in the list of three sets, the statement would be false, because one set would have no match. Please let me know if what I'm asking is unclear, and I can clarify.

• Generally, rather than finding some "formal-looking" notation for a complicated concept, it is better to just describe what you mean precisely in words. Commented Feb 6, 2021 at 20:12

In mathematics, a list is really a function from some indexing set - e.g. the list $$[\{1,2\},\{2\}, \{3,4\}, \{2\}]$$ (using Python notation) would correspond to the function with domain $$\{0,1,2,3\}$$ given by $$0\mapsto \{1,2\},\quad 1\mapsto \{2\},\quad 2\mapsto \{3,4\},\quad 3\mapsto\{2\}.$$ Per your comment to William Elliot, you're interested in the following property of a function $$F$$ whose codomain is the set of finite sets of natural numbers (I'm guessing based on your examples that you're not interested in non-natural numbers or infinite sets here):
There is a partition of $$dom(F)$$ into pairs such that for each pair $$\langle A,B\rangle$$ in the partition we have $$\sum_{x\in A}x=\sum_{y\in B}y$$.
(Note that this makes perfect sense regardless of what $$dom(F)$$ is; for example, it need not be ordered, or finite, or anything else nice.)