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In a set, the order of its elements is (as far as I know) not important; in a sequence, the order of its elements is important.

Which is the notation I should use in order to define a sequence? I could denote the sequence of positive integers by $S$.

Now, suppose I want to list (in order) the terms of sequence $S$. I'm not sure if the notation

$$S=\{1, 2, 3, 4, ...\}$$

is appropriate. Besides, I'm not sure if using $=$ is appropriate.

Any comments?

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    $\begingroup$ When you say "sequence of positive integers" do you mean the sequence $S=(s_i)_{i\geq 1}=(s_1,s_2,s_3,\ldots,s_i,\ldots)$ such that $s_i=i$ for all $i\geq 1$? Or, do you just mean the set $\{1,2,3,\ldots\}$. They are different things. $\endgroup$ – Dan Rust May 8 '14 at 21:18
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    $\begingroup$ The set $\{1,2,3,4,5\ldots\}$ is equal to $\{2,1,3,4,5,\ldots\}$. However, the sequence $(1,2,3,4,5\ldots)$ is not equal to $(2,1,3,4,5,\ldots)$. $\endgroup$ – Dan Rust May 8 '14 at 21:21
  • $\begingroup$ So basically you're saying that curly braces are used for sets and parentheses are used for sequences? $\endgroup$ – User X May 8 '14 at 21:23
  • $\begingroup$ Yep! And order matters for sequences but not for sets. $\endgroup$ – Dan Rust May 8 '14 at 21:24
  • $\begingroup$ Ok, so that means I could write $$S=(1,2,3,4,...)$$. Is this right? $\endgroup$ – User X May 8 '14 at 21:25
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You can denote the sequence by $$S = \{a_k\}_{k = 1}^\infty $$ where $a_k = k$.

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  • $\begingroup$ The braces $\{\}$ are confusing here. Normally we use brackets $()$ such as $S=(a_k)_{k=1}^{\infty}$. $\endgroup$ – Dan Rust May 8 '14 at 21:23
  • $\begingroup$ I use {} brackets though. $\endgroup$ – Anirban Ghosh May 8 '14 at 21:23
  • $\begingroup$ @DanielRust If one makes it clear one is talking about a sequence, then it should be fine. One might also talk about the sequence $\{(k,a_k):k=0,1,2,\ldots\}$, but then one is more succinct and write $a_k$ instead of $(k,a_k)$. $\endgroup$ – Pedro Tamaroff May 8 '14 at 22:11

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