Venn diagram
Venn diagrams graphical expressions used to show the relationship between two or more different sets of groups (Edwards, 2004). They show the logical relations among different sets indicating their similarities and differences (Edwards, 2004). For instance, if a class has 40 students with 25 students taking Accounting and 30 students taking Mathematics while five students are taking both Mathematics and Accounting, Venn diagrams can be used to show graphically the number of students taking each subject only as well those taking both subjects.
In her article, Amy Myers articulate that Venn diagrams are not limited to three or fewer sets. She argues that we can use Venn diagrams for problems with more than three sets (Myers, 2012). The concept of Venn diagrams involves using congruent circles to visualize the intersections that exist among two or more sets. One rule in the construction of Venn diagrams is that the circles must be congruent (Myers, 2012). This is pretty easy to achieve if there are three or fewer sets. When there are more than three sets, it is difficult to use congruent and show all the possible interactions among the different sets.
Myers provide that the most important thing in constructing a Venn diagram is to ensure that it shows all the possible intersections of the sets. The article explains the formula used for determining the number of possible intersections among sets (Myers, 2012). Myers explains that for a given number of sets, n, the number of all possible intersections is given by 2n. Thus, if there are four different sets, there will be 16 possible interactions between the sets.
She also explains how to determine the maximum number of regions that can be created with a given number of sets. To get the maximum number of regions, one should ensure that no more than two circles intersect at one point. The number of regions that can be created with n number of circles is given by n2 – n + 2. Given n number of sets, a Venn diagram must have 2n regions. Thus, it is not possible to create the required number of regions if the number of sets exceeds three.
Myers explored ways of constructing Venn diagrams if the number of sets is more than three. She concluded that it is not possible to draw Venn diagrams using circles, both congruent and non-congruent if the there are four or more sets. Quality Venn diagrams should have congruent shapes, the shapes should be convex, and the diagram should be symmetric (Myers, 2012). It is difficult to achieve all these qualities when the number of sets is more than three. However, these are just aesthetic qualities that may not interfere with the diagram’s ability to illustrate logical relationships between sets. Myers asks whether it is possible to draw Venn diagrams for all number of sets. According to John Venn, the inventor of Venn diagrams, it is possible to draw Venn diagrams for all number of sets. He further states that Venn diagrams can be drawn in any shapes and they don’t have to congruent, convex or have rotational symmetry. John Venn felt that the value of Venn diagrams depends on their purpose rather than the shape.
Myers states that Venn diagrams should be simple, congruent, convex and have rotational symmetry (Myers, 2012). She constructed Venn diagrams in different shapes other than circles. She concludes that Venn diagrams can be constructed for all values of n. However, it is impossible to achieve all the above qualities. Some can be congruent and have rotational symmetry but are not simple. Thus, any shape can be used so long it maintains some of the qualities and serves its purpose.
Lessons learned from the article
This article has been an eye-opener for me. I have always believed that Venn diagrams should only be drawn using congruent circles. Myers has enlightened me that we can use any shape to draw a Venn diagram provided the diagram has some of the qualities such as convexity, congruency, among other qualities. The most important thing is that the Venn diagram serves its purpose; that is, clearly showing the logical relations among different sets.
I also agree with Myers that the application of Venn diagrams cannot and should not be limited to three sets. The concept was introduced to help solve real life problems. There are cases where the number of groups or sets is more than three (Waddell Jr. & Quinn, 2011). Venn diagrams can still be used in such cases. However, circles may not be used in some cases, especially when the number of sets is large. The aesthetics or appearance of the Venn diagram is immaterial provided it serves the purpose. Venn diagrams for a large number of sets is complex. Due to the complexity, some software has been developed to help in the construction of Venn diagrams. This has helped in the application of Venn diagrams in complex situations involving several sets.
Through the article, I have learned formulas for determining the number of possible interactions among a given number of sets as well as the maximum number of regions. This also helped in understanding why drawing a Venn diagram using circles is difficult when eth number of sets is more than three. Several studies are undergoing on the concept of Venn diagrams. The article has been so valuable in enhancing my understanding of the concept.
References
Edwards, A. (2004). Cogwheels of the mind: the story of Venn diagrams (1st ed.). Baltimore:
Johns Hopkins University Press.
Myers, A. (2012). Are Venn Diagrams Limited to Three or Fewer Sets?. Department Of
Mathematics, Bryn Mawr Colleg: Pennsylvania, 1-10.
Waddell Jr., G. & Quinn, R. (2011). Two Applications of Venn Diagrams. Teaching
Statistics, 33(2), 46-48. http://dx.doi.org/10.1111/j.1467-9639.2010.00428.x
