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:
\( J\): \(\subseteq \mathbb{N}\)
\(*s\): \(\in \{\text{ the functions }\}\), denoted like \((s_1, s_2, ...)\) where \(s_j = s (J_j)\) where \(J_j\) is the \(j\)-th element of \(J\) ordered increasingly
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Conditions:
\(Dom (s) = J\)
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\(J\) is denoted as \(\{J_1, J_2, ...\}\) in the increasing order.
"\(n\)-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 \(J = \{2, 5, ...\}\), because otherwise, we would have to do an extra work of renumbering the index values just in order to call something "sequence", but as we often need to specify the \(j\)-th element of \(J\) but we cannot call it \(j \in J\) in general, we call it \(J_j \in J\).
\(\{J_1, ..., J_n\}\) is called "leading subset of \(J\)": while \(\{J_1, J_2\}\) is a leading subset, \(\{J_1, J_3\}\) is not any leading subset: in any sequence, the order is important, so, we often need to take a leading subset instead of just a subset.
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: \(Dom ((s_1, s_2)) = \{J_1, J_2\}\) while \(Dom ((s_1)) = \{J_1\}\).
