This is a continuation of discussion started in FINALLY: The Difference between Nerd, Dork, and Geek Explained by a Venn Diagram
Geeks and Nerds: a cartoon from XKCD (click on the image for a link to this wonderful site)
Venn diagrams works only for two or three sets, but not for four because 4 circles in the plane do not divide the plane in 16 regions. Venn diagrams are useful only because axioms of Boolean algebra can be written using only three variables, thus allowing for a diagrammatic representation of each axiom. I have made a similar observation elsewhere: Coxeter groups allow a powerful level of visualisation because all relations between their canonical generators are in some explicit mathematical sense two dimensional.
A later addition in response to a comment: If we do not insist on sets being represented by circles, then Venn diagrams (although increasingly non-intuitive) can be drawn for larger numbers of sets:
Symmetrical 5-set Venn diagram made congruent ellipses, by Branko Grünbaum
However, their educational value is limited.
And this is a link to Branko Grunbaum.
Mathematics under the Microscope 
Look at the venn diagram that divide a plane in 16 regions on the cover of this book: http://www.amazon.com/First-Course-Information-Theory-Technology/dp/0306467917
By: Adrian Tam on July 4, 2011
at 7:38 am
Thanks, it is an useful reminder.
By: Alexandre Borovik on July 4, 2011
at 8:30 am
Venn diagrams do have some educational value though. The
regions demarcated by the 
ellipses are the atoms in the free Boolean algebra on 
generators, and finite unions of those regions give all 
elements of the free Boolean algebra. It’s nice to have concrete pictures of formal facts like these.
By: Todd Trimble on July 4, 2011
at 10:35 am
Sorry, I had to say “Venn diagrams for more than three sets have no educational value at school level”
By: Alexandre Borovik on July 4, 2011
at 11:18 am
Venn’s novel contribution, and why we remember him still, was to include a boundary quadrilateral around all the circles, enabling visual representation of the complement of a set. No one had done this before him, and he did it at a time when there was great discussion and contention among statisticians about the appropriate definition of “a population” in undertaking statistical inference.
Personally, I would not call your second diagram a Venn diagram, since the diagram border is not explicitly drawn.
By: peter on July 7, 2011
at 9:28 pm
@peter: of course, you are right.
By: Alexandre Borovik on July 7, 2011
at 10:55 pm
Hi All,
There are related mathematics and algorithms problem here, for those who love mathematics / algorithm problem solviing Mathalon , after all whats in mathematics without problems.
-Hemant
By: Hemant Verma on July 10, 2011
at 7:44 am
Hemant asks “whats in mathematics without problems”. In fact, mathematics is the science of structure and pattern. Problems are a distraction from this, and from the subject’s main purpose. Let’s stop teaching math through problems, and let’s ditch the macho competitiveness that an unwarranted focus on problem-solving has led us to.
By: peter on July 10, 2011
at 12:28 pm
Peter, I agree with you entirely.
Unfortunately, Hemant is not asking anything. He’s spamming the comments section of the latest post of every math-oriented weblog he can find.
And he’s a terrible human being for it.
By: John Armstrong on July 10, 2011
at 3:53 pm