description/proof of that for linearly-ordered ring, finite number of subsets with same finite index set, and subset as sum of subsets with same index set, maximum of subset is equal to or smaller than sum of maximums of subsets and minimum of subset is equal to or larger than sum of minimums of subsets
Topics
About: ring
The table of contents of this article
Starting Context
- The reader knows a definition of linearly-ordered ring.
- The reader knows a definition of subset of linearly-ordered set with induced linear ordering.
- The reader knows a definition of maximum of partially-ordered set.
- The reader knows a definition of minimum of partially-ordered set.
- The reader admits the proposition that for any linearly-ordered set, any nonempty finite subset has the maximum and the minimum.
Target Context
- The reader will have a description and a proof of the proposition that for any linearly-ordered ring, any finite number of subsets with any same nonempty finite index set, and the subset as the sum of the subsets with the same index set, the maximum of the subset is equal to or smaller than the sum of the maximums of the subsets and the minimum of the subset is equal to or larger than the sum of the minimums of the subsets.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(R\): \(\in \{\text{ the linearly-ordered rings }\}\) with any linear ordering, \(\lt\)
\(J\): \(\in \{\text{ the nonempty finite index sets }\}\)
\(J'\): \(\in \{\text{ the nonempty finite index sets }\}\)
\(\{S_{j'} = \{s_{j', j} \in R \vert j \in J\} \vert j' \in J'\}\):
\(S\): \(= \{\sum_{j' \in J'} s_{j', j} \vert j \in J\}\)
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Statements:
\(Max (S) \le \sum_{j' \in J'} Max (S_{j'})\)
\(\land\)
\(\sum_{j' \in J'} Min (S_{j'}) \le Min (S)\)
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2: Note
\(Max (S)\), \(Max (S_{j'})\), \(Min (S_{j'})\), and \(Min (S)\) exist, by the proposition that for any linearly-ordered set, any nonempty finite subset has the maximum and the minimum.
This proposition requires \(J\) and \(J'\) be finite, because otherwise, \(\sum_{j' \in J'} s_{j', j}\) would not make sense in general and \(Max (S)\), \(Max (S_{j'})\), \(Min (S_{j'})\), and \(Min (S)\) would not exist in general.
2: Proof
Whole Strategy: Step 1: see that \(Max (S) \le \sum_{j' \in J'} Max (S_{j'})\); Step 2: see that \(\sum_{j' \in J'} Min (S_{j'}) \le Min (S)\).
Step 1:
\(Max (S) = \sum_{j' \in J'} s_{j', j}\) for a \(j \in J\).
But \(s_{j', j} \le Max (S_{j'})\) for each \(j' \in J'\).
So, \(\sum_{j' \in J'} s_{j', j} \le \sum_{j' \in J'} Max (S_{j'})\).
So, \(Max (S) \le \sum_{j' \in J'} Max (S_{j'})\).
Step 2:
\(Min (S) = \sum_{j' \in J'} s_{j', j}\) for a \(j \in J\).
But \(Min (S_{j'}) \le s_{j', j}\) for each \(j' \in J'\).
So, \(\sum_{j' \in J'} Min (S_{j'}) \le \sum_{j' \in J'} s_{j', j}\).
So, \(\sum_{j' \in J'} Min (S_{j'}) \le Min (S)\).
