Trying to combine limits of sumations I have the following situation. I want to simplify
$$ \sum_{P(i,j) \cap Q(i,j)} a_{ij} + \sum_{P(i,j) \cup Q(i,j)} a_{ij} $$
Where $P(i,j) = \{ 1 \leq i \leq j \leq n\} $ and $Q(i,j) = \{ 1 \leq j \leq i \leq n \} $.
try:
I know I can write $P(i,j) = \{1 \leq i \leq n \} \cap \{ 1 \leq j \leq n \} \cap \{ i \leq j \}$. Thus,
$$ P(i,j) \cap Q(i,j) = \{1 \leq i \leq n \} \cap \{ 1 \leq j \leq n \} \cap \{ i \leq j \} \cap \{1 \leq i \leq n \} \cap \{ 1 \leq j \leq n \} \cap \{ j \leq i \} = $$
$$ = \{1 \leq i \leq n \} \cap \{ 1 \leq j \leq n \} \cap \{ i \leq j \} \cap \{ j \leq i \} = $$
$$ = \{1 \leq i \leq n \} \cap \{ 1 \leq j \leq n \} \cap \{ i = j \} = \boxed{\{1 \leq i \leq n \} }$$
Is this correct? For the union I have
$$P(i,j) \cap Q(i,j) = \{1 \leq i \leq n \} \cap \{ 1 \leq j \leq n \} \cap \{ i \leq j \} {\color{red} \cup} \{1 \leq i \leq n \} \cap \{ 1 \leq j \leq n \} \cap \{ j \leq i \} = $$
$$ = ( \{1 \leq i \leq n \} \cap \{ 1 \leq j \leq n \} ) \cap (\{ i \leq j \} {\color{red} \cup}  \{ i \leq j \}) $$
But, I here Im stuck. Any help would be much appreciated.
 A: You're almost done here: you've reversed an $i \le j$ to $j \le i$, but aside from that mistake, you've gotten to $$P(i,j) \cup Q(i,j) = (\{1 \le i \le n\} \cap \{1 \le j \le n\}) \cap (\{i \le j\} \cup \{j \le i\})$$ and since one of $i \le j$ and $j \le i$ will always hold, this is just $\{1 \le i \le n\} \cap \{1 \le j \le n\}$.
(Also, in the first part, you dropped the $i=j$ condition. Instead of $1 \le i \le n$, you should have $1 \le i = j \le n$, because you do want to know what your constraints on $j$ are.)
But I think you're wrong to sort of think about this in terms of manipulating sets. You can see this quite clearly if you just think of $P(i,j)$ and $Q(i,j)$ as specifying subsets of the square grid of values $(i,j)$ where $1 \le i \le n$ and $1 \le j \le n$. For example, for $n=4$, we have 
$$P(i,j) = \left\{\begin{matrix}
    (1,1) & (1,2) & (1,3) & (1,4) \\
          & (2,2) & (2,3) & (2,4) \\
          &       & (3,3) & (3,4) \\
          &       &       & (4,4)
\end{matrix}\right\}$$
and
$$Q(i,j) = \left\{\begin{matrix}
    (1,1) &       &       &       \\
    (2,1) & (2,2) &       &       \\
    (3,1) & (3,2) & (3,3) &       \\
    (4,1) & (4,2) & (4,3) & (4,4)
\end{matrix}\right\}$$
from which it's easy to see that
$$P(i,j) \cap Q(i,j) = \left\{\begin{matrix}
    (1,1) &       &       &       \\
          & (2,2) &       &       \\
          &       & (3,3) &       \\
          &       &       & (4,4)
\end{matrix}\right\}$$
and 
$$P(i,j) \cup Q(i,j) = \left\{\begin{matrix}
    (1,1) & (1,2) & (1,3) & (1,4) \\
    (2,1) & (2,2) & (2,3) & (2,4) \\
    (3,1) & (3,2) & (3,3) & (3,4) \\
    (4,1) & (4,2) & (4,3) & (4,4)
\end{matrix}\right\}.$$
