How to add mathematical notation of a set

Question

I'm pretty new to Latex, so please bare with me.

I'm trying to add this notation to my document:

I created the above picture from the Online Latex Equation Editor, but it seems that when I try to apply the same code in my editor (TeXstudio), things are different...

For example, the above picture was produced using the code:

D=\left\{ x\in\mathbb{N}|1\leq x\leq 100 \right\} 

But when I write it in my editor (enclosed in the $ sign of course), I get the following error when compiling:

Undefined control sequence. $D=\left{x\in \mathbb

The funny thing is that I was still able to see the output in the preview window (even though the editor shows that this line has error in it).
That's the first problem.

The second problem is, although I can see the output in the preview window, it won't be the same as the output I get from the Online Latex Equation Editor.
While the Online Latex Equation Editor produces this:

My TeXstudio editor produces this:

Notice the distinction in the N symbol that denotes the set of natural numbers. It is different the what I intended, which is the first version, as it produced by the Online Latex Equation Editor.

What's wrong here? Am I missing a package or something?

Answer 1

A variant solution, also based on mathtools, with the cooperation of xparse allows for a syntax that's closer to mathematical writing: you just have to type something like\set{x\in E;P(x)} for the set-builder notation, or \set{x_i} for sets defined as lists.

Note that it's unnecessary to load amsmath if you load mathtools.

Illustration:

\documentclass[12pt]{article}
\usepackage[utf8]{inputenc}
\usepackage{amssymb}
\usepackage{lmodern}
\usepackage{mathtools, nccmath}
\newcommand{\N}{\mathbb N}
\newcommand{\Q}{\mathbb Q}

\usepackage{xparse}
%
\DeclarePairedDelimiterX{\set}[1]{\{}{\}}{\setargs{#1}}
\NewDocumentCommand{\setargs}{>{\SplitArgument{1}{;}}m}
{\setargsaux#1}
\NewDocumentCommand{\setargsaux}{mm}
{\IfNoValueTF{#2}{#1} {#1\,\delimsize|\,\mathopen{}#2}}%{#1\:;\:#2}

\parindent = 0pt

\begin{document}

 \[ D = \set{x ∈ \N ; 1\leq x\leq 100} \]%

The delimiters adjust to the size of the contents in the * version:
\[ E = \set*{x ∈ \Q ; -\frac{1}{2}\leq x \leq \frac{1}{2}}\]%

You also can have a manual adjustment with an optional argument to \verb+\set+:
\[ E = \set[\big]{x ∈ \Q ; -\mfrac{1}{2}\leq x \leq \mfrac{1}{2}}\]%

And you can define sets as simple lists:
\[ \text{Unit fractions}= \set*{\mfrac{1}{1}, \mfrac{1}{2},\mfrac{1}{3}, ... } \]%

 \end{document} 

Answer 2

Probably the online editor loads the required package, which is amssymb (recommended for documents with math material). Note that amsfonts could be sufficient, but amssymb provides a richer supply of symbols.

\documentclass{article}

\usepackage{amsmath}
\usepackage{amssymb}

\newcommand{\numberset}[1]{\mathbb{#1}} 
\newcommand{\nat}{\numberset{N}}}

\begin{document}

\[
D = \{\, x\in\nat \mid 1\le x\le 100 \,\}
\]

\end{document}

Note that \left and \right are not necessary here (and should be used only when really needed). Also the vertical bar should be \mid for better spacing. The thin spaces at both ends are a recommended, although optional, refinement.

The indirect definition of \nat (for “natural numbers”) is a trick that allows for uniform appearance of objects having the same nature. If you (or somebody else, read a fussy supervisor) later decide that boldface should be used, you can just change the definition of \numberset.

If you need plenty of set notations, it may be better to use mathtools:

\documentclass{article}

\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{mathtools}

\newcommand{\numberset}[1]{\mathbb{#1}}
\newcommand{\nat}{\numberset{N}}

\DeclarePairedDelimiterX{\Set}[2]\{\}{%
  \, #1 \;\delimsize\vert\; #2 \,
}

\begin{document}

\begin{gather*}
D=\Set{x\in\nat}{1\le x\le 100}\\
D=\Set[\big]{x\in\nat}{1\le x\le 100}\\
D=\Set[\Big]{x\in\nat}{1\le x\le 100}\\
D=\Set[\bigg]{x\in\nat}{1\le x\le 100}\\
D=\Set[\Bigg]{x\in\nat}{1\le x\le 100}\\
D=\Set*{x\in\nat}{1\le x\le \frac{200}{2}}
\end{gather*}

\end{document}

With the optional argument to \Set you can make the delimiters bigger. With \Set*, \left and \right are added.