definition of sequence
Topics
About: set
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
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:
\( \mathbb{N}\):
\( S\): \(\subseteq \mathbb{N}\)
\(*f\): \(\in \{\text{ the functions }\}\), denoted like \((e_1, e_2, ...)\) where \(e_j = f (l_j)\) where \(l_j\) is the \(j\)-th element of \(S\) ordered increasingly
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Conditions:
\(dom f = S\).
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2: Natural Language Description
For the natural numbers set, \(\mathbb{N}\), and any subset, \(S \subseteq \mathbb{N}\), any function, \(f\), such that \(dom f = S\), often denoted like \((e_1, e_2, ...)\) where \(e_j = f (l_j)\) where \(l_j\) is the \(j\)-th element of \(S\) ordered increasingly
3: Note
"sequence" often alludes to an infinite sequence, but at least this definition allows finite sequences.
While usually, \(S\) can be just like \(\{1, 2, ...\}\), the definition does not exclude cases like \(S = \{2, 5, 6\}\), because otherwise, we would have to do an extra work of renumbering the index values just in order to call something a sequence.
As a set, \(\{e_1, e_2\} = \{e_1\}\) when \(e_1 = e_2\), but as a sequence, \((e_1, e_2) \neq (e_1)\) even when \(e_1 = e_2\), because they are some different functions: \(dom (e_1, e_2) = \{1, 2\}\) while \(dom (e_1) = \{1\}\).
