# Sum notation of a tuple or set

I'm currently confused in how to express the sum of a tuple. I have a set or a tuple (for summation, order shouldn't be an issue) like this:

$$S_{A,B,C} = (1,3,6)$$

The subscript $A,B,C$ has nothing to do with the elements of the tuple, it identifies the exact type of S and is there to illustrate the variable naming.

I want to simply add all components: $1+3+6 = 10$.

My initial try was this:

$$X_{A,B,C} = \sum_{x \in S_{A,B,C}} x.$$

The next:

$$X_{A,B,C} = \sum_{x=1}^{|S_{A,B,C}|} S_{A,B,C,x}$$

But I was told that both versions are hard to read.

How would you recommend to write it?

• Why do you put equality between $S_{ABC}$ and your ideas of the sum notation? I thought that $S_{ABC}$ is the tuple and not its sum. – user87690 Nov 4 '14 at 10:25
• Note the commas. $S_{A,B,C}$ could mean that its a tuple and $S_{ABC}$ is the sum of it. So $S_{A,B,C} != S_{ABC}$. – Thomas Fankhauser Nov 4 '14 at 10:37
• Ok, but at least I missed the the difference when reading, so maybe it's prone to misread. – user87690 Nov 4 '14 at 10:46
• Thanks, you're right. It is pretty much misleading. I'll update it. – Thomas Fankhauser Nov 4 '14 at 11:11

## 1 Answer

I think that $X_{A, B, C} = ∑ S_{A, B, C} = ∑ (1, 3, 6) = 10$ works and is even formally correct. $∑_{i ∈ I} f(i)$ is a variant of $∑(f(i): i ∈ I) = ∑f$. The argument you give to a sum operator is a function rather than set, the same element can be there more times. And an $n$-tuple is a function from a set of form $\{1, …, n\}$ or $\{0, …, n - 1\}$ depending on convention.

• Okay thanks, that's much easier, yes. – Thomas Fankhauser Nov 4 '14 at 11:12