Venn diagrams are useful since they represent every possible intersection between the sets that one is studying. Suppose that we want to draw a Venn diagram for a single set. Then it is enough to draw it in zero dimensions (the zero-dimensional space is just the space with one point): the set is represented by the single point in the zero-dimensional space. If we want to draw a Venn diagram that represents two sets we notice that the zero-dimensional space won’t do, there is not enough space. But in the one-dimensional space (the real line) we can draw both sets and in such a way that we have distinct regions for , and . Once again if we want to draw one more set, one dimension won’t do. But we can draw the classical Venn diagram in two dimensions.

*First exercise:* Show that we cannot draw a Venn diagram in two dimension for four sets using only circles.

*Second exercise:* Show that this can be done if we use for example rectangles.

*Third exercise*: Can we draw a Venn diagram for four sets in two dimensions in such a way that the region that represents every intersection is connected? And simply connected?

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HINT: some graphs are not planar.

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