Ordering differences between ordered numbers

Question

Consider 2 sets of numbers $\{a_1,...,a_J\}$ and $\{b_1,...,b_J\}$.

(A1) We know that $a_1<...<a_J$ and $b_1<...<b_J$.

Now consider the $J^2$ differences of each element of the first set with each element of the second set: $$ a_1-b_1\\ a_1-b_2\\ ...\\ a_1-b_J\\ a_2-b_1\\ a_2-b_2\\ ...\\ a_2-b_J\\ ...\\ a_J-b_J $$

Does (A1) allow to order from smallest to largest the differences? If not, does (A1) allow to say which is the biggest and the smallest difference? If not, does (A1) allow to say anything about the ordering of the differences?

(suggestions on more appropriate tags are welcome!)

Answer 1

If you arrange the differences $a_i-b_j$ in an $J\times J$ grid so that the $(i,j)$ entry is $a_i-b_j$, then the rows will increases from left to right and the columns will increase from top to bottom. If an entry $x$ is to the right of and below an entry $y$, then you know $x\ge y$. However, this is all you know about the ordering. For any entries $z$ and $w$ where $z$ is strictly to the right of and strictly above $w$, you do not know the ordering of $z$ and $w$.

In particular, the smallest entry is $a_1-b_J$, and the largest is $a_J-b_1$.