Use curly brackets for sets

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In a recent assignment, I saw a student who used parentheses (round brackets) to represent a set: \begin{equation} (1, 2, 3, \cdots). \end{equation} Of course, as Georg Cator put it, "the essence of mathematics lies in its freedom," so it is OK to use whatever symbols to represent a set, if you define it first, consistently. However, using parentheses for sets is unconventional. We conventionally use curly brackets instead: \begin{equation}\{1, 2, 3, \cdots \}\end{equation} I recommend you stick to this convention to avoid any confusion.

By the way, a list of elements enclosed by parentheses, such as in \((1, 2, 3)\), is often used to represent a tuple. A tuple is an ordered list of elements. As such, \((1, 2, 3)\) and \((2, 1, 3)\), for example, represent different tuples. By comparison, in sets, the order of elements does not matter so that \(\{1, 2, 3\}\) and \(\{2, 1, 3\}\) represent the same, identical set. Furthermore, redundant elements are allowed in tuples. That is, \((1, 1, 2)\) is a valid tuple, whereas \(\{1, 1, 2\}\) is not a valid set because the two "1" are not distinguishable (or it should be treated as \(\{1, 2\}\) by removing the redundancy).

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