How to write a set with an index

Question

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?

Answer 1

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

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

Answer 2

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 $I=\{1,2,...n\}$

Answer 3

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.