While carrying out a task, the following question arose.
Suppose we have two sets of positive numbers, $C_1$ and $C_2$ respectively. In which it is fulfilled:
- $C_1$ and $C_2$ have an equivalent order, example: $C_1 = \lbrace 1, 10, 20, 40 \rbrace$ and $C_2 = \lbrace 90, 1000, 5000, 10000 \rbrace$. The equivalent of 1 in $C_2$ is 90, the equivalent of 10 in $C_2$ is 1000, etc. The important thing is that 1 < 10 and, equivalently, 90 < 1000 and so on with $\textbf{all elements and their relationships} $
- $|C_1| = |C_2|$
- In both $C_1$ and $C_2$ there are no repeated elements
Let $S_1^{m}$ and $S_2^{m}$ be the $\textbf{sum sets}$ of $C_1$ and $C_2$ that have $m$ elements respectively. Given the two sets of sums, let's say they are ordered. That is to say: if $C_1 = \lbrace 1, 10, 20, 40 \rbrace$ and $C_2 = \lbrace 90, 1000, 5000, 10000 \rbrace$ and $m=3$ then $S_1^m = \{ 31 , 51, 70\}$ and $S_2^m = \{ 6090, 11090, 16000 \}$.
Let $s_i$ and $s_j$ be a particular sum in $S_1^m$ and $S_2^m$ (i and j are the indices that correspond to each sum in the respective sum set).
In this context, I would like to know: if $s_i$ and $s_j$ take elements at the same indices of each corresponding set then $i=j$, that is, the partial sums that take elements at the same indices of the sets are located at the same index in the respective sets of partial sums even though the elements of the sum are different.
It should be noted that I am not talking about the literal sum, that is, if the elements add 500, 1094, etc. but the equivalence between orders of particular numbers is transferred to their sums and combinations.
Please, if someone could answer me, I would be very grateful since it has been on my mind and seems simple.