I'm self studying linear algebra from the book by L.Mersky and struggling with the below theorem.
If $(λ1,....λn),(μ1,...μn)$ and $(K1,....Kn)$ are arrangements of $(1,....n)$ then 1.
The proof starts by saying that if $(λ1,...λn)$ and $(μ1,...μn)$ are subjected to the same derangement,then the value of the left hand side of the above equation remains unaltered. To prove this we observe that $$(λk_j- λk_i)(μk_j- μk_i) = (λs-λr)(μs-μr)$$ where $r=\min(k_i,k_j)$ and $s=\max(k_i,k_j)$
If $r,s$(such that $1\leq r < s \leq n$) are given,then there exist unique integers $i,j$ (such that $1\leq i < j \leq n$) satisfying $r=\min(k_i,k_j)$ and $S=\max(K_i,K_j)$. Thus there exist biunique correspondence between the pairs $K_i,K_j$ and the pairs $r,s$. Hence,..(the multiplicative function and signum function is applied to both sides of the above observation line and proved the theorem)
Can somebody please explain what is the meaning of $r=\min(k_i,k_j)$ and $s=\max(k_i,k_j)$?