Do any 3D Venn diagrams exist that have no 2D analogues?

[SamRoss]

TL;DR

Can any relationships between sets be dreamed up in a 3D Venn diagram that could not be done in a 2D Venn diagram?

If we were to use three-dimensional spheres to represent sets, could a 3D Venn diagram be constructed that could not be drawn as a normal 2D Venn diagram without changing the relationships between the sets?

 

[Stephen Tashi]

Summary:: Can any relationships between sets be dreamed up in a 3D Venn diagram that could not be done in a 2D Venn diagram?

What's your definition for a "relationship between sets"? If two sets have a unique single element in common, is that considered a relationship?

 

[pbuk]

Summary:: Can any relationships between sets be dreamed up in a 3D Venn diagram that could not be done in a 2D Venn diagram?

If we were to use three-dimensional spheres to represent sets, could a 3D Venn diagram be constructed that could not be drawn as a normal 2D Venn diagram without changing the relationships between the sets?

You should be able to work this out for yourself. I can think of a structure that can be represented in a 2D Venn diagram that cannot be properly represented using 3-spheres: A = {1, 2}, B = {1, 3}, C = {1, 4}.

Spoiler

The three spheres must all intersect because $1 \in A \cap B \cap C$ but if this is the case then the volume $A \cap B \cap C'$ will not be empty.

Edit: why did you decide to restrict your 3D Venn diagram to 3-spheres: 2D Venn diagrams are not restricted to circles - actually shape doesn't really mean anything in the spaces in which we 'draw' Venn diagrams?

 

[phinds]

Venn diagrams only require closed spaces so 2D does not have to be just circles and 3D does not have to be just spheres. Without the circle/sphere restrictions, it might always be possible to do a 3D that equates to a 2D, but I don't know that for sure.