Drawing labeled intervals with specific patterns using TikZ

I am trying to draw a diagram using TikZ in which I can specify particular elements within the interval that satisfy a certain condition. This example seems to be the "closest" to what I intend to draw. I tried to modify it, however, due to my limited skills in TikZ, I wasn't able to specify the pattern of labeling, and to draw the interval end braces properly. Any suggestions or hints shall be appreciated. Here's a try.

Update: Changing the value of \bound will change the maximum number of points on each side of N. Changing the value of the x key will change the distance between adjacent points. (Thanks to @Jubobs for fueling this change)

Code

\documentclass[border=2pt]{standalone}
\usepackage{tikz,amssymb}
\usetikzlibrary{calc}

\newcommand\bound{10} % bound number of points on each side of N
\newcommand\labelnum[]{
\begin{scope}[font=\footnotesize,x=.3cm,#1]
\foreach \mypt in {0,#2,...,\bound}{
}
\draw(-\bound-5,0)--(\bound+5,0) node[pos=0,left]{$\mathbb Z$};
\node(start)[at={(-\bound-4,0)},label=below:{$N-d$}]{$[$};
\node(end)[at={(\bound+4,0)},label=below:{$N+d$}]{$]$};
\node[at={($(start)!.5!(end)$)},label=below:{$N$}]{\vphantom{$[$}};
\node[at={(-\bound-2,0)},above]{$\cdots$};
\node[at={(\bound+2,0)},above]{$\cdots$};
\node[at={(0,0)},above=5pt]{#3};
\end{scope}
}

\begin{document}
\begin{tikzpicture}
\labelnum{1}{All integers are labeled}
\labelnum[yshift=-2cm]{2}{Even numbers are labeled}
\labelnum[yshift=-4cm]{5}{Multiples of 5 are labeled}
\labelnum[yshift=-6cm]{3}{Multiples of 3 are labeled}
\end{tikzpicture}
\end{document}

Output • You shoudn't have so many "magical numbers". – jub0bs Jan 22 '14 at 23:45
• @Jubobs: I've made an update. Does it look better now? :) – Herr K. Jan 23 '14 at 0:03
• I shall now bestow a +1 on you :). – jub0bs Jan 23 '14 at 0:17
• Thank you. This certainly matches what I wanted to draw. Is there an easy way to add the labels, e.g., "Multiples of 3 are labeled"? – Abdullah Heyari Jan 23 '14 at 13:49
• @AbdullahHeyari: See updated answer. I modified \labelnum to take an extra argument, which sets the "title" of the line. – Herr K. Jan 23 '14 at 17:30

A PSTricks solution where the diagram has been 'automated'; one can now choose all the absolute and relative distances in the diagram and it will automatically change accordingly.

\documentclass{article}

\usepackage{amssymb}
\newcommand*\setZ{\mathbb{Z}}

\usepackage{expl3}
\ExplSyntaxOn
\cs_new_eq:NN \calc \fp_eval:n
\ExplSyntaxOff
% values needed (automatically calculated)
\newcommand*\StartDiagram{\calc{\Beginning-0.4}}
\newcommand*\EndDiagram{\End}
\newcommand*\HeightAxis{\calc{(#1-1)*\AxesDist+0.35}}
\newcommand*\HeightDrawing{\calc{\HeightAxis{3}+0.35}}
\newcommand*\StartDist{\calc{\Beginning+\StartIndent} }
\newcommand*\StartDots{\calc{\StartDist+1+\StartDistIndent}}
\newcommand*\WidthDots{\calc{\End-\Beginning-2*(\StartIndent+1+\StartDistIndent)}}
\newcommand*\DistDots{\calc{\WidthDots/(#1-1)}}
\newcommand*\SmallDotsStart{\calc{\StartDist+\SmallDotsIndent}}
\newcommand*\SmallDotsEnd{\calc{\End-(\StartIndent+1)}}
\newcommand*\SmallDotsDist{\calc{(1-\SmallDotsIndent)/2}}
\newcommand*\Middle{\calc{(\Beginning+\End)/2}}

%%% parameters %%%
% endpoints of axes
\newcommand*\Beginning{2 }
\newcommand*\End{12 }
% distance between axes
\newcommand*\AxesDist{2.5}
% constants of indentation on axes
\newcommand*\StartIndent{0.75}
\newcommand*\SmallDotsIndent{0.5}
\newcommand*\StartDistIndent{0.25}
% number of big, `hollow' dots on the lines
\newcommand*\DotsLineTop{23}
\newcommand*\DotsLineMiddle{7}
\newcommand*\DotsLineBottom{5}

\begin{document}

\begin{pspicture}(\StartDiagram,0)(\EndDiagram,\HeightDrawing)
{\psset{dotstyle = Bo}
\multido{\rC = \StartDots+\DistDots{\DotsLineTop}}{\DotsLineTop}{\psdot(\rC,\HeightAxis{3})}
\uput(\Middle,\HeightAxis{3}){\scriptsize Even numbers are labeled}
\multido{\rB = \StartDots+\DistDots{\DotsLineMiddle}}{\DotsLineMiddle}{\psdot(\rB,\HeightAxis{2})}
\uput(\Middle,\HeightAxis{2}){\scriptsize Multiples of $3$ are labeled}
\multido{\rA = \StartDots+\DistDots{\DotsLineBottom}}{\DotsLineBottom}{\psdot(\rA,\HeightAxis{1})}
\uput(\Middle,\HeightAxis{1}){\scriptsize Multiples of $5$ are labeled}}
\multido{\r = 0.35+\AxesDist}{3}{%
\psline(\Beginning,\r)(\End,\r)
\uput(\Beginning,\r){$\setZ$}
\multido{\rD = \SmallDotsStart+\SmallDotsDist, \rE = \SmallDotsEnd+\SmallDotsDist}{3}{%
{\psset{dotsize = 2pt}
\psdot(!\rD\space \r\space 0.1 add)
\psdot(!\rE\space \r\space 0.1 add)}}
\psdot(\Middle,\r)
\uput(\StartDist,\r){\scriptsize $N-d$}
\uput(\Middle,\r){\scriptsize $N$}
\uput(!\Beginning \End add \StartDist sub \r){\scriptsize $N+d$}
\rput(!\Beginning \End add \StartDist sub \r\space 0.1 add){$]$}
\rput(!\StartDist \r\space 0.1 add){$[$}}
\end{pspicture}

\end{document} • Thank you. Your suggested answer introduced me to the power of PSTricks! – Abdullah Heyari Jan 23 '14 at 13:51
• @AbdullahHeyari Glad I could help. :) (Remember to accept one of the answers in due time.) – Svend Tveskæg Jan 23 '14 at 13:59
• I shall select an answer soon. Something is hindering the proper execution of the PSTricks on my computer. I received multiple error messages saying: Undefined control sequence \multido – Abdullah Heyari Jan 23 '14 at 15:49
• @AbdullahHeyari Have you got an updated LaTeX distribution? If not, try to update it. – Svend Tveskæg Jan 23 '14 at 16:00

Another solution with PSTricks just for spending the boring time.

\documentclass[pstricks,border=12pt,12pt]{standalone}
\usepackage{multido,amssymb}
\SpecialCoor
\psset{xunit=2.5mm,yunit=5mm}

\begin{document}
\begin{pspicture}(-30,-9)(30,1)
\foreach \y/\t in {0/{Even numbers },-4/{Multiples of 3 },-8/{Multiples of 5 }}
{
\psline(-28,\y)(30,\y)
\psset{labelsep=10pt}
\uput(0,\y){\t are labeled}
\uput[-90](0,\y){$N$}
\uput(-22,\y){$\cdots$}
\uput(22,\y){$\cdots$}
\rput(-25,\y){$[$}
\rput(25,\y){$]$}
\uput[-90](-25,\y){$N-d$}
\uput[-90](25,\y){$N+d$}
\uput{6pt}(-28,\y){$\mathbb{Z}$}
}
\multido{\i=-20+1}{41}
{
\pscircle*[linecolor=red](!\i\space dup 2 mod 0 ne {/SD {} def} if 0){2pt}
\pscircle*[linecolor=green](!\i\space dup 3 mod 0 ne {/SD {} def} if -4){2pt}
\pscircle*[linecolor=blue](!\i\space dup 5 mod 0 ne {/SD {} def} if -8){2pt}
}
\end{pspicture}
\end{document} 