I'm recently trying to understand maths more deeply, and I've always struggled with more complicated equations featuring multiple sum symbols. Particularly, I never can tell with exactity what is under the influence of a sum symbol. I usually determine this using context. Is there a way to explicitly state what is and isn't still under the sum influence?
For example, in the inclusion-exclusion principle from Set theory:
$$\biggl|\bigcup_{i=1}^n A_i\biggr|=\sum_{i=1}^n\left|A_i\right|-\sum_{i < j}\left|A_i\cap A_j\right|\>\>+\sum_{i < j < k}\left|A_i\cap A_j\cap A_k\right|-\ \cdots\ + \left(-1\right)^{n+1} \left|A_1\cap\cdots\cap A_n\right|$$
Is the second sum under the first one? Is the third sum under the second one? Is it at the same time still under the first one? Is this somehow implicitly indicated in this equation? Is there any explicit way of indicating this?
Best Answer
A commonly (though not universally) followed convention is that the scope of a summation is broken by addition or subtraction (but not multiplication). So if you write something like $$\sum_i a_i +1$$ that means $$\left(\sum_i a_i\right)+1$$ and not $$\sum_i(a_i+1).$$ Similarly, in your example, the summations would be broken up by the addition and subtraction between them, so they are not nested.
Note that in contrast, if there are multiplications between summations, then the multiplication is evaluated inside the summation and so the sums are nested. For instance, $$\sum_i a_i\sum_{j>i} b_{ij}$$ is commonly written with the meaning $$\sum_i \left(a_i \sum_{j>i} b_{ij}\right)$$ (and so the inner sum is a sum over values of $j$, not values of the pair $(i,j)$, since the $i$ is bound instead by the outer sum).
That said, these conventions are not universal and are typically not written down and taught explicitly, so you will probably encounter some deviations from them. If you are ever worried about whether the intended meaning is clear in something you're writing, it's always safe to insert parentheses to clarify the meaning (as I did in the examples above).