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.
Best Answer
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.