In a finite set, is there a concept/notation for the smallest value larger than a particular element? For example, I have a sorted set as $ A = \{a_1, a_2, ..., a_k \} $ where $ a_2 > a_1, a_3 > a_2, ... , a_{k} > a_{k-1}, $. Given an arbitrary value $p$ not be in the set, I want to find the smallest element $a_k$ in the set greater than $p$. Any idea?
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1$\begingroup$ Not following. Why isn't it just $a_{m+1}$? $\endgroup$– luluMay 19, 2022 at 16:35
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3$\begingroup$ Maybe the notation you're looking for is $\min\{a\in A:a>x\}$. $\endgroup$– KarlMay 19, 2022 at 16:41
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$\begingroup$ @lulu Edited the question $\endgroup$– ewr3243May 19, 2022 at 16:44
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$\begingroup$ You use $a_m$ to denote the $m^{th}$ element in the set and also to denote some value which may or may not be in the set? That is extremely poor notation. $\endgroup$– luluMay 19, 2022 at 16:54
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$\begingroup$ The term "successor" might be appropriate. Its meaning should be defined in the context, of course. $\endgroup$– Greg MartinMay 19, 2022 at 17:28
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As @Karl mentioned in the comments, $\min\{a\in A:a>x\}$ might be what you want. Also, for finite sets, you can also use infimum (or $\inf$). However, $\inf$ only gives the minimum for finite sets and some infinite sets like $[0,1]$ - for other infinite sets, it is the greatest lower bound. See this Wikipedia page to learn more: https://en.wikipedia.org/wiki/Infimum_and_supremum
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$\begingroup$ I was aware of infimum and supremum. However how can they be used with respect to a particular value, such as $p$ described in the question? I think they are absolute for a set, but you can explain in the answer if I am missing anything. $\endgroup$– ewr3243May 19, 2022 at 19:54
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$\begingroup$ @ewr3243 Hello again. How about $\inf\{a\in A:a>x\}$? $\endgroup$– MathGeekMay 20, 2022 at 11:40