I am reading Donald Knuth's Concrete Mathematics (2nd Edition) and I am on chapter 2 (Sums).
I have problems in understanding his some notations on multiple sums. I quote his explanations I can't understand from his book.
$$\sum_{j\in J}\sum_{k\in K(j)}a_{j,k}=\sum_{k\in K'}\sum_{j\in J'(k)}a_{j,k}$$ Here the sets $J, K(j), K'$, and $J'(k)$ must be related in such a way that $$[j\in J][k\in K(j)]=[k\in K'][j\in J'(k)]$$ A factorization like this is always possible in principle, because we can let $J=K'$ be the set of all integers and $K(j)=J'(k)$ be the basic property $P(j,k)$ that governs a double sum.
My questions are:
- What is $K$ and $K(j)$? Is $K$ a function and $K(j)$ is the range of $K$?
- Why the equivalence of $J$ and $K'$ is a set and the equivalence $K(j)$ and $J'(k)$ is a property?
- So, if $J$ and $K'$ are sets and $K$ and $J'$ are functions, what will be $J, K(j), K'$ and $J'(k)$ in this case? $$[1\le j\le n][j\le k\le n]=[1\le k\le n][1\le j\le k]$$