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I have two types of sets, one of them is a list of points of the form $\{t_{k}\}$ and the other one is an interval of the form $[a,b]$. And my custom set consists of these type of sets and I need to plot its Cartesian product. As an example, say my set is $A=[0,1]\cup\{2,3,4,5\}$, I need to plot $A\times A$. If I have only list of points, then I have no problem but when intervals come in to play, I don't have a very good idea. I need help at this point.

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up vote 5 down vote accepted

Try the following:

For example, $A=[0,1]\cup\{2,3,4\}\cup[5,7]$.

points = {2, 3, 4};
intervals = {{0, 1}, {5, 7}};

Graphics[{Lighter@Blue, Point@Tuples[points, 2], 
  Rectangle @@ Transpose[#] & /@ Tuples[intervals, 2], 
  Line[{{{#1, #2[[1]]}, {#1, #2[[2]]}}, {{#2[[1]], #1}, {#2[[2]], #1}}}] & @@@ 
     Tuples[{points, intervals}]}, Axes -> True]

enter image description here

The disadvantage of the considered method is different width of lines and points. Adjusting the PointSize and Thickness does not help.

Let us consider another method and $A=[0,1]\cup\{3/2,4/3,\dots,(n+1)/n,\ldots\}$.

points = Table[(n + 1)/n, {n, 2, 30}];
intervals = {{0, Min[points]}};

I use Min[points] instead of 1 to remove the gap between interval and finite number of points.

min = Min[points, intervals] - 0.01;
max = Max[points, intervals] + 0.01;
size = 1000;
thickness = 2;
data = ConstantArray[0, size];
scale = Round@Rescale[#, {min, max}, {1, size}] &;

Here 0.01 is a small space around the data, size is the size of partitioning, and thickness measured in the integers units.

(data[[scale[#1] ;; scale[#2]]] = 1) & @@@ intervals;
(data[[scale[#] - thickness ;; scale[#] + thickness]] = 1) & /@ points;
img = ColorNegate@Image@Outer[Times, data, data, 1 - {0.33, 0.33, 1}];

Here {0.33, 0.33, 1} is color (light blue).

  Polygon[{{min, min}, {min, max}, {max, max}, {max, min}}, 
   VertexTextureCoordinates -> {{0, 1}, {0, 0}, {1, 0}, {1, 1}}]}, 
 Axes -> True]

enter image description here

share|improve this answer
How can we make width of lines and dots same size? Also, what would you suggest for $A=[0,1]\cup\{3/2,4/3,\cdots,(n+1)/n,\cdots\}$? – bkarpuz Sep 24 '13 at 21:04
@bkarpuz It is the interesting question. See my update. – ybeltukov Sep 24 '13 at 23:04
I got the idea for filling the gap between the interval and the "cut" sequence. How about we use another set $B$ to fill the gap and take Cartesian product of $A\cup B$. For instance $B=\{1,31/30\}$ as a filler interval? But it seems that setting the size of the points and the width of the lines is the main issue. – bkarpuz Sep 25 '13 at 5:57

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