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?