# Which is the best notation for a sequence?

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.

• 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 8 '14 at 21:18
• 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 8 '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 8 '14 at 21:23
• Yep! And order matters for sequences but not for sets. – Dan Rust May 8 '14 at 21:24
• Ok, so that means I could write $$S=(1,2,3,4,...)$$. Is this right? – User X May 8 '14 at 21:25

You can denote the sequence by $$S = \{a_k\}_{k = 1}^\infty$$ where $a_k = k$.
• The braces $\{\}$ are confusing here. Normally we use brackets $()$ such as $S=(a_k)_{k=1}^{\infty}$. – Dan Rust May 8 '14 at 21:23
• @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 Tamaroff May 8 '14 at 22:11