507: Sequence

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definition of sequence

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Topics

About: set

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of natural numbers set.
• The reader knows a definition of function.

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Target Context

• The reader will have a definition of sequence.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$\mathbb{N}$:
$S$: $\subseteq \mathbb{N}$
$\ast f$: $\in \{\text{ the functions }\}$, 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:
$domf=S$.
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"$n$-sequence" means any sequence whose domain has the cardinality $n$.

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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 $domf=S$, 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

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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, $(e_1,e_2)\ne (e_1)$ even when $e_1=e_2$, because they are some different functions: $dom(e_1,e_2)=\{1,2\}$ while $dom(e_1)=\{1\}$.

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References

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