$\begingroup$Given a set of numbers $S=\{x_1,\dotsc,x_{|S|}\}$, where $|S|$ is the size of the set, what would be the appropriate notation for the sum of this set of numbers? Is it
$$\sum_{x_i \in S} x_i \qquad\text{or}\qquad \sum_{i=1}^{|S|} x_i$$ or something else?
$\endgroup$13 Answers
$\begingroup$I'd write either $\displaystyle\quad\sum_{i=1}^{|S|} x_i\quad$ or $\displaystyle\quad\sum_{x\in S} x$.
If the second form is used, then the subscript is just clutter.
Some mathematicians (perhaps especially set theorists?) might write $\displaystyle \sum S$. This would parallel the way set theorists write $\displaystyle \bigcup S$ where others might write $\displaystyle \bigcup_{x\in S} x$.
$\endgroup$8$\begingroup$Both expressions are acceptable with the second being more usual in this context.
The expression $$\sum_{x \in S} x$$ is more common when $S$ is implicitly defined, e.g., when one is summing over all prime numbers. The expression $$\sum_{i =1}^{|S|} x_i$$ would be more common here because you are explicitly given the list of elements of the set $S$.
$\endgroup$1$\begingroup$Say I had a set A, under an operation with the properties of $+$, then $$\sum_{i\in A} x_i$$ is how I write it.
$\endgroup$2
