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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?

User X
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    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. – Dan Rust May 08 '14 at 21:18
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    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 $(2,1,3,4,5,\ldots)$. – Dan Rust May 08 '14 at 21:21
  • So basically you're saying that curly braces are used for sets and parentheses are used for sequences? – User X May 08 '14 at 21:23
  • Yep! And order matters for sequences but not for sets. – Dan Rust May 08 '14 at 21:24
  • Ok, so that means I could write $$S=(1,2,3,4,...)$$. Is this right? – User X May 08 '14 at 21:25
  • If you specifically want the elements to be in that order, then yes. If the order matters, or you want to repeat elements (such as $(1,1,2,2,3,3,4,4\ldots)$) then stick to sequences. If you don't care about order or knowing if any elements have been repeated, then use sets. – Dan Rust May 08 '14 at 21:27
  • Thank you again! Now suppose that $(19,20)$, $(29,30)$, $(39,40)$ are three sequences that form another sequence, which we will denote by $T$. How do you express the fact that $T$ is made up of those three sequences in a specific order? Would it be correct to write $$T=((19,20),(29,30),(39,40))$$? – User X May 08 '14 at 21:57
  • In other words, are the expressions $T=((19,20),(29,30),(39,40))$ and $T=(19,20,29,30,39,40)$ equivalent? – User X May 08 '14 at 22:08
  • A sequence has infinitely many component elements, we normally call an ordered finite set of elements a 'tuple'. You can think of an infinite tuple as being a sequence. – Dan Rust May 08 '14 at 22:08
  • So I guess I can think that a sequence (which has infinitely many component elements) is made up of infinitely many tuples. So this means that $T=((19,20),(29,30),(39,40))$ is the same as $T=(19,20,29,30,39,40)$. Am I right? – User X May 08 '14 at 22:16
  • Not really, two sequences $s=(s_i){i\geq 0}=(s_0,s_1,s_2,\ldots)$ and $t=(t_i){i\geq 0}=(t_0,t_1,t_2,\ldots)$ are defined to be equal if and only if $s_i=t_i$ for all $i\geq 0$. It is similar for tuples. In your case, for the first you have $s_0=(19,20)$ but for the second you have $t_0=19$ and so they are not equal. – Dan Rust May 08 '14 at 22:22
  • I see, you're right. I think I'm not asking the right question. Maybe I'll ask my question later on. – User X May 08 '14 at 22:32
  • I would suggest just googling the specific definitions of sets and sequences and what it means for them to be equal. If you have not taken a course which introduces you to set theory notation, it may be a steep curve due to the abstraction involved (specifically sets of sets). – Dan Rust May 08 '14 at 22:38
  • I think this is what I'm looking for (concatenation of sequences). – User X May 09 '14 at 02:30
  • According to the comments here, it seems that the expressions $T=(19,20,29,30,39,40)$ and $T=(19,20)^\frown (29,30)^\frown (39,40)$ are equivalent, but I'm not sure if the symbol $^\frown$ is widely known and/or accepted. – User X May 09 '14 at 02:38

1 Answers1

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You can denote the sequence by $$S = \{a_k\}_{k = 1}^\infty $$ where $a_k = k$.

aghost
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  • The braces ${}$ are confusing here. Normally we use brackets $()$ such as $S=(a_k)_{k=1}^{\infty}$. – Dan Rust May 08 '14 at 21:23
  • I use {} brackets though. – aghost May 08 '14 at 21:23
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    @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 $(k,a_k)$. – Pedro May 08 '14 at 22:11