definition of sequence
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of natural numbers set.
- The reader knows a definition of function.
Target Context
- The reader will have a definition 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:
\( \mathbb{N}\):
\( J\): \(\subseteq \mathbb{N}\)
\(*s\): \(\in \{\text{ the functions }\}\), denoted like \((s_1, s_2, ...)\) where \(s_j = s (l_j)\) where \(l_j\) is the \(j\)-th element of \(J\) ordered increasingly
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Conditions:
\(Dom (s) = J\).
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"\(n\)-sequence" means any sequence whose domain has the cardinality \(n\).
2: Note
"sequence" often alludes to an infinite sequence, but at least this definition allows finite sequences.
While usually, \(J\) can be just like \(\{1, 2, ...\}\), the definition does not exclude cases like \(J = \{2, 5, 6\}\), because otherwise, we would have to do an extra work of renumbering the index values just in order to call something "sequence".
As a set, \(\{s_1, s_2\} = \{s_1\}\) when \(s_1 = s_2\), but as a sequence, \((s_1, s_2) \neq (s_1)\) even when \(s_1 = s_2\), because they are some different functions: \(Dom ((f_1, f_2)) = \{1, 2\}\) while \(Dom ((f_1)) = \{1\}\).
