# How to write a set with an index

I'd like to write a set $\{x_1, x_2, ..., x_n\}$ in a simple way.

What is a popular way?

In my high school, I wrote it as $\{x_i\}_{i=1}^{n}$. Is it a correct way?

• This normally denotes a sequence, which is not entirely the same thing as a set. I'd argue that $\{x_1,\ldots,x_n\}$ is a simple way to write a set. Jun 29, 2016 at 13:25
• $(x_1,\ldots ,x_n)$ denotes an $n$-tuple, not a set. You could write $\{x_i\mid i\in I\}$, for $I=\{1,\ldots ,n\}$. Jun 29, 2016 at 13:25
• Just writing $\{x_i\}$ is not unusual, provided $n$ is clear from context. Jun 29, 2016 at 13:38

What you described is not a set, you probably mean $\{{x_1,x_2...,x_n}\}$

This is usually denoted as ${A=\{{x_i|i\in I}\}}$ Where $I$ is your index set $I=\{1,2,...n\}$

For the set $\{x_1,\ldots,x_n\}$ you can write

$$\begin{cases}\{x_i\}_{i=1}^n \\ \\ \{x_i , 1\leq i\leq n\} \\ \\ \{x_i\}_{1\leq i\leq n}. \end{cases}$$

in the set theory we have $\{x_1,...,x_n\}=\{x_i\mid 1\leq i\leq n\}=\cup_{1\leq i\leq n}\{x_i\}$ so is the set of $n$ elements, and the set $\{x_i\}^{n}_{i=1}=\{x_1\}\{x_2\}\cdot\cdot\cdot\{x_n\}=\{(x_1,\cdot\cdot\cdot x_n)\}$ is a set of one element.

• rkjt50r983, the crucial point that all of these people are trying to make is the difference between a "set", in which order is not relevant, and a "sequence", in which order is relevant. Is this a set or a sequence? If it is not a sequence, why are the entries indexed like that? Jun 29, 2016 at 15:09
• in my opinion the symbol $\{...\}$ is reserved for the set, and so $\{x_i\}^{n}_{i=1}$ is a set as we write $\{x\}^m$, but the symbol for a sequence as one element is $(.)$, and so the element $(x_i)_{1\leq i\leq n}$ is a sequece.thanks Jun 29, 2016 at 15:49
• Actually, $x_i$ is the i-th data in large data set. I have a lot of docments (e.g. news articles) and x_i is the i-th document in the large document set (e.g. collection of news articles). Jun 29, 2016 at 23:16