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I do not have a background in mathematics. However, I do empirical research and I oftentimes have difficulties finding the correct mathematical notations for certain things I want to express:

I am comparing two sets of values. Each value in one set relates to a certain value in the other set, i.e. the two sets represent a set of "pairs". How do I express that, e.g. more than half of the values in one set are larger than their respective counterpart in the other set?

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Let the two sets be $A$ and $B$; the wording implies that they are finite. You say that all the elements can be paired, so there is a bijective function $f:A\mapsto B$ expressing this relation. Call the cardinality of each set $n$.

Your last sentence may then be written as $$|\{a\in A:a>f(a)\}|>\frac n2$$

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  • $\begingroup$ Does a similar notation hold for vectors? For example, if A and B are vectors, does the notation still hold? $\endgroup$ – shenflow Apr 20 '20 at 10:00
  • $\begingroup$ @shenflow I suppose so, if you regarded vectors as ordered multisets. $\endgroup$ – Parcly Taxel Apr 20 '20 at 10:04
  • $\begingroup$ Alright. So using a different notation for vectors, one could say $f: \vec{A} \mapsto \vec{B}$, etc.....? Thank you in advance. $\endgroup$ – shenflow Apr 20 '20 at 10:12
  • $\begingroup$ @shenflow I would put $f:\mathbb R^n\to\mathbb R^n$ or something similar. $\endgroup$ – Parcly Taxel Apr 20 '20 at 10:13
  • $\begingroup$ Okay. So you would put that. Then say that the two vectors are elements of the domain and codomain respectively. And then say $$|\{a\in A:a>f(a)\}|>\frac n2$$ to express that more than half of the elements in A are larger than their counterpart in B? Sorry if this gets to extensive, maybe I should post this as a question on its own. $\endgroup$ – shenflow Apr 20 '20 at 10:18

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