definition of union of subsequences of sequence
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of subsequence of sequence.
Target Context
- The reader will have a definition of union of subsequences of sequence.
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:
\( s\): \(\in \{\text{ the sequences }\}\), with \(Dom (s) = J\) and \(Ran (s) = S\)
\( L\): \(\in \{\text{ the index sets }\}\)
\( \{s^`_l = s \circ f_l: J^`_l \to S \in \{\text{ the subsequences of } s\} \vert l \in L\}\)
\(*\cup_{l \in L} s^`_l\): \(\in \{\text{ the subsequences of } s\}\), \(= s \circ id_{\cup_{l \in L} f_l (J^`_l)}\), where \(id_{\cup_{l \in L} f_l (J^`_l)}: \cup_{l \in L} f_l (J^`_l) \to J\) is the identity map
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Conditions:
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2: Note
This definition should be the one that many people should naturally imagine.
In fact, \(\cup_{l \in L} s^`_l\) is better understood as \(\cup_{l \in L} \widetilde{s^`_l}\) where \(\widetilde{s^`_l}\) is the canonical subsequence for \(s^`_l\): how each \(f_l: J^`_l \to J\) is taken is not important and taking \(\cup_{l \in L} J^`_l\) does not make sense, so, we take the canonical subsequences and take the union of the canonical subsequences, and if the domain of \(\cup_{l \in L} s^`_l\) needs to be tweaked, we can take any appropriate \(f: J^` \to \cup_{l \in L} f_l (J^`_l)\).
\(\cup_{l \in L} s^`_l\) is indeed a subsequence, because \(\forall f_m (j^`_{m, 1}), f_n (j^`_{n, 2}) \in \cup_{l \in L} f_l (J^`_l) \text{ such that } f_m (j^`_{m, 1}) \lt f_n (j^`_{n, 2}) (id_{\cup_{l \in L} f_l (J^`_l)} (f_m (j^`_{m, 1})) = f_m (j^`_{m, 1}) \lt f_n (j^`_{n, 2}) = id_{\cup_{l \in L} f_l (J^`_l)} (f_n (j^`_{n, 2}))) \land \forall j \in J (\exists f_m (j^`_m) \in \cup_{l \in L} f_l (J^`_l) (j \le f_m (j^`_m)))\).
For example, for a sequence, \(s\), with \(Dom (s) = \mathbb{N}\), the subsequence, \(s^`_1\), with \(Dom (s^`_1) = \mathbb{N}\) and \(f_1 (\mathbb{N}) = \{0, 1, 20, 21, ...\}\), and the subsequence, \(s^`_2\), with \(Dom (s^`_2) = \mathbb{N}\) and \(f_2 (\mathbb{N}) = \{10, 11, 30, 31, ...\}\), taking like \(f (0) = 0, f (1) = 10, f (2) = 1, f (3) = 11, ...\) does not create any subsequence, because the order is disturbed, so, we take \(f_1 (\mathbb{N}) \cup f_2 (\mathbb{N}) = \{0, 1, 10, 11, 20, 21, 30, 31, ...\}\) and \(id_{f_1 (\mathbb{N}) \cup f_2 (\mathbb{N})}\).
