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Let $X=\{X_1,\dots,X_n\}$ be a finite collection of finite sets $X_i$. Let \begin{gather} \prod_{i\in N}X_i \end{gather} be its Cartesian product. I have seen an authoritative reference in my field denote an element of this Cartesian product as \begin{gather} \{x_i\}_{i\in N}\in\prod_{i\in N}X_i \end{gather} I am somewhat confused, as an element in the Cartesian product is a vector, and I thus think that the following notation is more natural: \begin{gather} (x_i)_{i\in N}\in\prod_{i\in N}X_i \end{gather} I thought that the use of “$()$” to denote vectors (ordered tuples) and the use of “$\{\}$” to denote sets (non-ordered tuples) was commonplace, but now I doubt it.

Can anyone please tell me whether I should use “$()$” or “$\{\}$” to denote an element of a Cartesian product?

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    $\begingroup$ Cartesian products are ordered and your assessment is perfectly valid. Parens[( )] are ordered tuples while braces[{ }] are unordered, and unique, hence sets. So, parens are the way for CP! $\endgroup$ Commented Jan 3, 2023 at 20:34
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    $\begingroup$ While I agree with the comment of @RaviTiwari, nonetheless when reading mathematical works one has to use common sense in understanding notations, because mathematicians use all sorts of alternative notations all the time. $\endgroup$
    – Lee Mosher
    Commented Jan 3, 2023 at 20:38
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    $\begingroup$ I agree with the OP and the comments above. However, there's a long-standing tradition of denoting sequences (clearly ordered) with curly braces that contradicts this, and these are cartesian products of sets indexed over the natural numbers. (I know, you only consider finite collections here.) $\endgroup$ Commented Jan 3, 2023 at 20:39
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    $\begingroup$ $\{x_i\}_{i=1}^n$ is usually notation for a sequence. There is no real logical difference between a sequence and an ordered tuple, but there is a semantic difference, and we tend to use tuples written as $(x_i)_{i=1}^n$ for the product of sets. In particular, sequences tend to come from an individual global set (a sequence in $X,$ say.) While tuples are not usually bound that way. $\endgroup$ Commented Jan 3, 2023 at 20:40
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    $\begingroup$ Different authors prefer different notations. E.g. in the magnificent book General Topology, by R. Engelking. $\endgroup$ Commented Jan 4, 2023 at 0:26

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