3.1.4: Visualising sets
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A Venn diagram shows all possible logical relations between a finite collection of different sets. These diagrams depict elements as points in the plane, and sets as regions inside closed curves. So a Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set. The points inside a curve (circle) labelled S represent elements of the set S, while points outside the boundary represent elements not in the set S. Figure 4.1 shows our example set which opened the section.
Venn diagrams help us to visualise sets and set operations. For example, the set of all elements that are members of both sets S and T, S ∩ T, is represented visually by the
Figure 4.3: Venn diagram of the intersection of two sets.
area of overlap of the regions S and T: see Figure 4.3. In Venn diagrams the curves are overlapped in every possible way, showing all possible relations between the sets. You can find it useful to draw a Venn diagram to gain intuition of what’s happening. On their own, Venn diagrams do not offer a proof for theorems in set theory however.