Quick questions about summation notation used in my book

Question

My Questions

1. In the screenshot below, is the summation notation in the orange (top) box and blue (bottom) box the exact same? So in a $3$ by $3$ matrix you'd only want to sum $a_{12} + a_{13} + a_{23}$.
2. Is it correct to say that the orange box is exactly the same as $$\sum_{i = 1}^{n-1} \sum_{j = i + 1}^n P(A_i A_j)?$$
3. Is it correct to say that the BLUE box is exactly the same as $$\sum_{(i,j):i<j} E[\min(X_i,X_j)]?$$
4. Why would you choose one notation or the other? It seems the blue box is faster to write.
5. Is it correct to say that the following will always hold if $i$ and $j$ span the exact same indices and $a_{ij} = a_{ji}$ for all $i$ and $j$? $$\sum_{i \neq j} a_{ij} = 2\sum_{i<j} a_{ij}$$ Thank you for your time and patience.

---

Answer 1

I will answer your questions one by one.

1. Your interpretation is correct.
2. Yes.
3. Yes.
4. It depends, if the author is trying to emphasize the minimum property or if he is trying to emphasize what motivated the minimum property.
5. Correct again!

Great work making sure you really understand everything in the book!