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=2^{4} 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 `n`^{2}-`n`+2 regions (see plane division by circles for details). This is well short of the 2^{n} "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 2^{n} 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.