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Interestingly, Venn diagrams are completely inadequate for their purpose: it's bad enough that you can't understand anything with a Venn diagram of 4 circles; what's worse is that you only get 14 regions, rather than the 16=24 you'd need!

This is the plane division by circles problem (see MathWorld with that name, which also gives sequence A014206 of Sloane's On-Line Encyclopedia of Integer Sequences). n≥1 circles divide the plane into at most n2-n+2 regions (see plane division by circles for details). This is well short of the 2n "regions" into which n sets can divide their domain. Still, the first 3 values are indeed 2,4,8, which leads to the common misconception that the sequence "correctly" continues 16,32,...

Thinking about it, it seems plausible that you'll have difficulty dividing the plane into 2n regions using only simple paths...

Venn diagrams might be neat for drawing on the blackboard, but only if you've got at most 3 sets.