description/proof of that for product set, subset, subset that contains 1st subset, and element of subproduct, cross section of 1st subset is contained in cross section of 2nd subset
Topics
About: set
The table of contents of this article
Starting Context
Target Context
- The reader will have a description and a proof of the proposition that for any product set, any subset, any subset that contains the 1st subset, and any element of any subproduct, the cross section of the 1st subset by the element is contained in the cross section of the 2nd subset by the element.
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:
\(J'\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\(\{S_{j'} \in \{\text{ the sets }\} \vert j' \in J'\}\):
\(\times_{j' \in J'} S_{j'}\): \(= \text{ the product set }\)
\(Q_1\): \(\subseteq \times_{j' \in J'} S_{j'}\)
\(Q_2\): \(\subseteq \times_{j' \in J'} S_{j'}\)
\(J\): \(\subset J'\), such that \(J \neq \emptyset\)
\(\times_{j \in J} s_j\): \(\in \times_{j \in J} S_j\)
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Statements:
\(Q_1 \subseteq Q_2\)
\(\implies\)
\({Q_1}_{[\times_{j \in J} s_j]} \subseteq {Q_2}_{[\times_{j \in J} s_j]}\)
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2: Proof
Whole Strategy: Step 1: see that for each \(\times_{l \in J' \setminus J} s_l \in {Q_1}_{[\times_{j \in J} s_j]}\), \(\times_{l \in J' \setminus J} s_l \in {Q_2}_{[\times_{j \in J} s_j]}\).
Step 1:
Let \(\times_{l \in J' \setminus J} s_l \in {Q_1}_{[\times_{j \in J} s_j]}\) be any.
\(\times_{j' \in J'} s_{j'} \in Q_1 \subseteq Q_2\).
So, \(\times_{l \in J' \setminus J} s_l \in {Q_2}_{[\times_{j \in J} s_j]}\).
