Based on sorting data using a Database Management System (DBMS), first perform different sorting operations on different columns, then execute opposite sorting operations on different columns compared to the former, verifying that the former and latter are reverse sequences of each other.
Problem: For the same dataset, SORT(ID1 DESC, ID2 ASC, ID3 DESC) and SORT(ID1 ASC, ID2 DESC, ID3 ASC) are inverse sequences of each other. Here, SORT() represents the sorting method used in database management systems, DESC indicates descending order, and ASC indicates ascending order.
Proof: Let P(x,y,z) be the proposition that 'SORT(ID1 DESC, ID2 ASC, ID3 DESC) and SORT(ID1 ASC, ID2 DESC, ID3 ASC) are inverse sequences of each other.' Each piece of data in the dataset to be sorted contains three elements, defined as (x_i, y_i, z_i), where i ∈ [1, n] represents the row number of each data entry. The three data elements in each row form a unit with invariance, following the sorting rules for columns x, y, z in a specific order.
(1) When there are no duplicate data elements in the dataset
Assume SORT[DESC(x_1), ASC(y_1), DESC(z_1)] = SORT[DESC(x_n), ASC(y_n), DESC(z_n)]. During the sorting process, following the sorting rules for columns x, y, z and based on the core invariance of data elements in the same row, we can deduce:
SORT[DESC(x_i), ASC(y_i), DESC(z_i)] = SORT[DESC(x_(n-i+1)), ASC(y_(n-i+1)), DESC(z_(n-i+1))]
By iterating cyclically, we prove that P(x, y, z) holds.
(2) When each row in the dataset contains duplicate data elements in different positions
When there are duplicate data elements in the x-column data, the x-column elements can be divided into duplicate and non-duplicate parts. Based on the row settings of the x-column, the y-column conforms to the conclusion in (1) within the range of non-duplicate rows in the x-column; therefore, P(x, y, z) holds.
Based on the core invariance of data elements in the same row, for the y-column within the range of duplicate data element rows in the x-column at the same position, the range of the duplicate part is [i, i+x], where x ∈ [0, n-i]. Then, the sortable data elements in the y-column are the part from y_i to y_(i+x), and this part follows the y-column sorting rules. After sorting, we can deduce:
SORT[DESC(x_i), ASC(y_i)] = SORT[DESC(x_(i+x)), ASC(y_(i+x))]
By iterating cyclically, we prove that P(x, y) holds.
When the x-column and y-column contain duplicate data elements, based on the above conclusions for the z-column, assuming the range of rows with duplicate data elements in the y-column is [i, i+y], where y ∈ [0, x], then the sortable data in the z-column is the part from z_i to z_(i+y), and this part follows the z-column sorting rules. After sorting, we can deduce:
SORT[DESC(x_i), ASC(y_i), DESC(z_i)] = SORT[DESC(x_(i+y)), ASC(y_(i+y)), DESC(z_(i+y))]
By iterating cyclically, we prove that P(x, y, z) holds.
From (1) and (2), we can see that the above proposition holds.