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Just a quick question for which I've yet to managed to find an answer using Google.

Given a set of numbers $S=\{x_1,\ldots,x_{|S|}\}$, where $|S|$ is the size of the set, what would be the appropriate notation for the sum of this set of nubmers?

Is it

$$\sum_{x_i \in S} x_i$$ or $$\sum_{i=1}^{|S|} x_i$$ or something else?

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up vote 14 down vote accepted

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 $\displaystyle \bigcup_{x\in S} x$.

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The set theorist's notation is useful, because it does not invoke dummy variables. The idea is that $\sum$ should be a function of type $\mathscr{P} \mathbb{R} \to \mathbb{R}$ (or replace $\mathbb{R}$ with your favourite ring). This idea eventually leads to the general notion higher-order functions and functors and monads... – Zhen Lin Feb 21 '12 at 17:46

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 $S$.

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As Michael Hardy says below, the subscripts are unnecessary and unsightly in the first expression. – Tom Cooney Feb 21 '12 at 17:15

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