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I was exploring the meaning of sequence, and that what I found

Let $$ be a non-empty set.

A sequence of elements of $$ is map $$ from $ℕ$ to $$.

That is, the domain of $$ is the whole set $\mathbb{N}$

$$ : ℕ \to $$

where $_:=()∈$, for all $∈ℕ$.

Recall that $ℕ={1,2,3,…}$

So, my question is, can elements of sequences be treated like pairs? where the first element of the pair is the order and the second element is the value.

Greg Nisbet
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Ahmad Shaaban
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    Yes, you could certainly look at them in that sense. In particular, a sequence is a $\textit{function}$ from the naturals into your given ambient space. And the first component of your "ordered pairs" will start at 1 and go up by 1 each time. For example the sequence ${\frac{1}{n}}_{n \in \mathbb{N}}$ can be viewed as $(1,1),(2,\frac{1}{2}),...$. hope this helps – homosapien Sep 06 '21 at 21:27
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    ^ they can (because of one of the common definitions of a function), but usually, operationally, people don't think in this way about functions or sequences, so if you do so, make sure to be very explicit. – peek-a-boo Sep 06 '21 at 21:31
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    Please use MathJax to formulate the mathematics in your MSE questions. As for your a question, a standard representation of a sequence in set theory is as a set of pairs. If you are trying to exploit this outside set theory, I recommend you think carefully about what you are trying to achieve. since your readers may not be comfortable with this kind of representation. – Rob Arthan Sep 06 '21 at 21:31
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    To help get you started, I formatted your post with block quotes (introduced using >) and added some simple mathjax. For future reference, \mathbb{N} is the natural numbers and \to is a function arrow. Also, please include a reference to where you are taking this definition from if possible. – Greg Nisbet Sep 06 '21 at 21:34

2 Answers2

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The answer is yes if you consider the simple example of a function that generates Pell numbers by iteration. Pairs of these values are needed for Euclid's formula $$ \qquad A=m^2-k^2\qquad B=2mk \qquad C=m^2+k^2\qquad$$

to generate Pythagorean triples where $\quad A-B=\pm1. \space$

\begin{equation} \text{The formula}\quad \quad m=k+\sqrt{2k^2+(-1)^k} \quad\text{generates}\quad 1,2,5,12,29,70,\cdots \end{equation}

Each $\space n^{th}\space\space k$-value generates an $m$-value and each $m$-value becomes the $\space (n+1)^{th} \space\space k$-value.

\begin{align*} k=1\qquad &\implies m=(1+\sqrt{2(1)^2+(-1)^1}\space)\big)=2\space & F(2,1)=(3,4,5)\\ k=2\qquad &\implies m=(2+\sqrt{2(2)^2+(-1)^2}\space)\big)=5\space & F(5,2)=(21,20,29)\\ k=5\qquad &\implies m=(5+\sqrt{2(5)^2+(-1)^5}\space)\big)=12\space & F(12,5)=(119,120,169) \end{align*}

Every adjacent pair corresponds to an $\space n^{th}\space$ Pythagorean triple where $\space n\in\mathbb{N}.$

poetasis
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In set theory, it is common to define a function $f$ as any set of ordered pairs such that if $(a,b)\in f$, and $(a,c)\in f$, then $b=c$. Then, we can define a sequence as a function with a domain of $\Bbb N$. So the answer to your question is yes, we can think of elements of sequences as ordered pairs.

However, many mathematicians would feel uncomfortable with the idea that a sequence is literally just a set of ordered pairs satisfying certain properties. Arguably, the set theoretic construction of sequences only exists to demonstrate that they can be given a formal definition, meaning that we can feel confident that the everyday notion of sequences doesn't lead to any sort of contradiction. But in practice, it is seldom useful to think of mathematical objects as literally being the same as their set theoretic representations. For further discussion of this philosophical point, see here.

Joe
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