# Chapter 5: Venn diagrams Versus Euler Diagrams

Venn diagrams present the relationship between two or more things with clarity and simplicity. From experience, I feel that memorizing and understanding relationships become much easier with the aid of a Venn diagram. However, it is important to understand how Venn diagrams are drawn, especially when there are more than three sets involved. The diagrams below contain four circles but they are slightly different from each other in terms of the shape of the circles and the number of overlapping areas.

According to the Collins English Dictionary, a Venn diagram also known as a set diagram is “a diagram in which mathematical sets or terms of a categorial statement are represented by overlapping circles within a boundary representing the universal set, so that all possible combinations of the relevant properties are represented by the various distinct areas in the diagram.” As shown above, Figure 1 does not have any overlapping areas that cover only blue and red as well as only yellow and green. Therefore, according to the definition, Figure 1 is not a Venn diagram since it does not show all the possible combinations that should exist when comparing four sets. On the other hand, Figure 2 does show all the combinations, thus making it a Venn diagram.

“If Figure 1 is not a Venn diagram, then what is it?” This question was rolling around in my head, but was answered when I came across a diagram called the Euler diagram. The Euler diagram was invented by Leonhard Euler, a Swiss Mathematician and a physicist, to show the relationship between two or more things. From this explanation, Venn diagrams and Euler diagrams can be interpreted as the same diagram. However, the key difference between these two diagrams is that an Euler diagram only shows the relationships that exist, while a Venn diagram shows all the possible relationships, meaning that it even depicts the relationships that are unfeasible. The subset and disjoint in Figure 3 and 4 will help assist in understanding the difference between the two.

The unfeasible area in the Venn diagram represents the null set which is a set that is empty

As shown above, in order to present a subset and a disjoint in a Venn diagram, the overlapping areas without any elements also need to be presented. This is because Venn diagrams present all combinations even if they are unfeasible. If not, then it is an Euler diagram, because Euler diagrams only show the relationships that exist.