# Summation with multiple variables seprated by comma.

Suppose there are two summations.

1)$\sum_{i=1}^na_i$

2) $\sum_{i=1}^nb_i$

Both summations are same but have different variables. Instead of writing both separately, Is it possible to merge them into single summation? Where variables are separated by comma. (Comma being considered OR)

For example is it correct to merge the above 2 summations in either of following way.

Merging summation 1 and 2.

$\sum_{i=1}^n(a_i, b_i)$ where comma serves the purpose of OR operation.

OR

$\sum_{i=1}^n(a, b)_i$ where comma serves the purpose of OR operation.

I have analytical model of time in a censor network system (Based on Summation). I have created another analytical model of energy analysis of that CNS. Both are quite lengthy and take a lot of space. Energy model is hundred percent replica of Time model but have different variable E (energy), instead of T(time). Writing both separately could be redundancy because both model are almost same just variable is different, I am figuring out how to merge them so one summation could be use to describe both variables time and energy. It is up to the reader which variable concerns him. So that why I am asking to separate them with OR operator.

• Maybe if I had a physics background the context would make it very clear to me what you mean, but speaking only for myself I would be very confused by both $(a_i, b_i)$ and $(a, b)_i$. When you say "OR operation," do you mean Boolean or bitwise? Aug 24 '18 at 15:18
• Sir, basically its not physics, its computer science. If we just put aside the background and talk about math, is there any way to merge two summations which are not related to each other? Above described in the question. Take it like this, There are two math model based on summations. Booth are replicas of each other just one variable is different in both of them. Instead of writing both separately I am looking for some way to merge those two because the summations are same for both just variable is different. Aug 25 '18 at 5:32
• Okay, let's say $a = \{1, 2, 3\}$ and $b = \{3, 2, 1\}$. Would the merged sum then be 7? Or would it be 12? Or something else entirely? Aug 25 '18 at 5:38
• (Summation is just a sum of variable that is in front of it) sir not merging the each element. My understanding of summation, is it like loop to staring index to last index. In Summation "i" is indexing variable, Consider it a loop "i" to "n", then both summation has same loop condition "i" to "n". Taking about computer programing if we have this kind of situation we can use single loop instead of using different loops for each variable. (Definitely it is possible in programming). I have mostly programming background so I am wondering that if it is possible in programming it might be in math. Aug 26 '18 at 16:57
• @RobertSoupe Sir I am not talking about merging of sets or elements. I am talking about merging condition for both variables. Because both variables of summation has same condition "i" to "n". As above in the question I have tried to merged the same condition for both "a" and "b". So my question is simple in math is there any way to merge summations based on condition and still keep the variables separated to each other. Aug 26 '18 at 16:59

It is permissible to add or subtract summations when the indices start and end at the same values. So you can do this: $$\sum_{i=1}^{n}a_i\pm\sum_{i=1}^{n}b_i=\sum_{i=1}^n(a_i\pm b_i).$$ Multiplications are possible, but not so easy: you have to change one of the sum variables to avoid confusion: $$\left(\sum_{i=1}^na_i\right)\left(\sum_{i=1}^nb_i\right)= \left(\sum_{i=1}^na_i\right)\left(\sum_{j=1}^nb_j\right)=\sum_{i=1}^n\sum_{j=1}^na_ib_j.$$ This works because of the distributive law. Division I wouldn't even worry about. The AND or OR operations would both be possible if the $a_i$ and $b_i$ are boolean variables - otherwise what would be the meaning of it? In that case you would have to use the distributive laws of boolean algebra carefully, and you would have to define what the "sum" meant.
• @AhmadBilal: In that case, I would recommend a linear combination of the models: $$\sum_{i=1}^n (\alpha a_i+\beta b_i),$$ with $\alpha$ and $\beta$ arbitrary. That gives you the flexibility to do whatever. Aug 24 '18 at 14:17