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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 $s=\max(k_i,k_j)$?

Thank you.

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    I'm not entirely sure if this helps you but min(number1, number2) is just the minimum of the two numbers i.e. the smaller one. Similarly max(number1, number2) is the maximum i.e. the bigger of the two numbers.2012-11-23
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    It would be great if you could at least clarify what's going on in the proof. You define $(x_1,\ \cdots,\ x_n)$ and $(y_1,\ \cdots,\ y_n)$ but neither seem to be used in your image. What does the $\varepsilon$ denote?2012-11-23
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    @ EuYu-sorry I made a mistake.Both are same.I just edited it.2012-11-23
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    This remains a bit unclear. Is this a proof about the sign of a permutation? Would it be possible to include an image of the page in question perhaps? Does it look a bit like the first page of this [pdf](http://www.math.mcgill.ca/goren/AlgebraII/permutations.pdf)?2012-11-23

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