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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?

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    Knuth's Concrete Mathematics p. 22 discusses this, and has a whole chapter about the summation operator.2017-05-31

3 Answers 3

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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...2012-02-21
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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.2012-02-21
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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.

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    You'd be summing undefined items2017-12-22