In chapter 2 section 4 (multiple sums) of Concrete Mathematics(Graham,Knuth,Patashnik) the authors use Iverson Identity to rearrange the variables' bounds. In particular, they start off with a question like this:
\begin{equation} S = \displaystyle\sum\limits_{1 \le j < k \le n}^{}{(a_k - a_j)(b_k - b_j)} \end{equation}
Continuing by changing the variable to get the lower triangle (from the diagonal): \begin{equation} S = \displaystyle\sum\limits_{1 \le k < j \le n}^{}{(a_j - a_k)(b_j - b_k)} \end{equation}
Then they go on to add S to itself using the Iverson identity: $$[1 \le j < k \le n] + [1 \le k < j \le n] = [1 \le j, k \le n] - [1 \le j = k \le n]$$
Here is where I feel absolutely lost. I can't understand how they arrive from the two inequalities in the left to the first in the right? On the left, both $k$ and $j$ have lower bound of 1, yet in the first on the right the $k$ is not bounded and $j$ appear not to have an upper bound. What am I missing?
Basically, what is meant there is:
$$\{(j,k): 1\leq j < k \leq n \text{ or } 1\leq k < j \leq n\}$$ $$ = \{(j,k): 1\leq j, k \leq n \} \setminus \{(j,k): 1\leq j= k \leq n\}$$