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How can I represent a set of distinct numbers in shorthand mathematical notation? In particular I am trying to write that there are 4 distinct integers $a,b,c,d$.

For three numbers, I could say:

$a,b,c \in \mathbb{Z} :a \not= b\not= c \not=a$; however, for larger sets, I need to compare every pair of numbers for a total of $\frac{n!}{2}$comparisons.

Is there an accepted notation for distinctness? Or could I make a statement such as $a,b,c,d \in \mathbb{Z}, \{a,b,c,d\}$ is a set, bearing in mind that sets cannot have duplicates?

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    $\begingroup$ Use words. "Let $A = \{ \ldots \}$ be a set of distinct integers." Say "$k$ distinct integers" if that will help your reader. Avoid confusing shorthand. $\endgroup$ Feb 10, 2018 at 0:04
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    $\begingroup$ The easiest way is to use words! Just say "Let $a,b,c,d$ be distinct integers." There is no shame in using the English language (or whichever) rather than using mathematical symbols in order to convey information. Another convenient option would be in labeling these with subindices and referring to arbitrary indices, as in "Let $a_1,a_2,a_3,a_4,\dots, a_n$ where $a_i\neq a_j$ for all $i\neq j$." $\endgroup$
    – JMoravitz
    Feb 10, 2018 at 0:04
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    $\begingroup$ sets cannot have duplicates It is generally accepted that $\,\{a,a\}=\{a\}\,$. Just spell it out in words "distinct integers", there is no good reason to make it more complicated than needs be. $\endgroup$
    – dxiv
    Feb 10, 2018 at 0:06

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Hint: Recall that both representations \begin{align*} \{1,1,2,3\}=\{1,2,3\} \end{align*} are valid and denote the same set. So this is not the appropriate notation for your needs.

It's common to use the phrase pairwise distinct for instance

  • Let $a,b,c,d\in\mathbb{Z}$ denote pairwise distinct integers which ...

The idea behind this phrase is that whenever we select two of the elements under consideration then they are different.

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