Venn s Diagrams: Maximum sets drawing

Transcription

1 Venn s Diagrams: Maximum sets drawing Aureny Magaly Uc Miam @academ01.ccm.itesm.mx ITESM Campus Ciudad de México ABSTRACT In this paper, I tried to describe the maximum number of sets which can be draw with Venn s Diagrams. KEYWORDS Sets, Venn s Diagrams. INTRODUCTION Venn diagrams were introduced in 1880 by John Venn, "M.A. Fellow and Lecturer in Moral Science, Caius College, Cambridge University", in a paper entitled On the Diagrammatic and Mechanical Representation of Propositions and Reasonings which appeared in the Philosophical Magazine and Journal of Science S. 5. Vol. 9. No. 59. July 1880, [1]. The diagrams popularly are associated with Venn, in fact, they are originated much earlier. Fig. 1 B. Representation of Venn s Diagrams We can know the number of sets in Venn n Diagrams whit the equation 2 where n is the number of sets represented by Venn Diagrams. B1. Using circles That way we can say: Venn Diagrams for 2 set. 2 n = 2 2 = 4 Fig.2 Venn Diagrams for 3 set. 2 n = 2 3 = 8 Fig.3 A. What is a Venn s Diagram? A Venn diagram is an illustration of the relationships between and among sets, groups of objects that share something in common. These illustrations are graphs that generally are circles, but not necessary; also closed other curves. Venn diagrams are used to depict set intersections. Fig. 2 Fig. 3 1

2 Whit n=4 there are an especial situation; if we can try draw a Venn s Diagram, is possible that we obtain something like Fig. 4, but is recommendable that we draw it with ellipses Fig. 5, because in some cases some areas would be missing. Fig. 4 Fig. 6 Fig. 5 B2. Using Ellipses Until now, we can see Venn s Diagrams for n=1, 2, 3, 4. Now, with n=5 we can obtain that the relations are 32 (2 5 =32). This is represented by Fig. 6. In Fig. 7 we can observed a Venn s Diagrams whit ellipses but the difference with Fig.6 is that the Fig.7 is a Symmetric Venn s Diagrams. Symmetric Venn s Diagrams is called when is used congruent ellipses or curves Fig. 7 and the diagram has a very a pleasing symmetry, namely an n-fold rotational symmetry. This simply means that there is a point x about which the diagrams may be rotated by 2 i pi / n and remain invariant, for i=0, 1,..., n-1. But, for n=5 is possible represented for other figures like squares or rectangles. Fig. 8 and Fig. 9 are examples for. 2

3 B3. Other representations Exist another figures which we can represented Venn s Diagrams like triangles, squares, and others figures and these could be symmetric or not. The follow figures are some example from these. Fig. 8 Fig. 10 Fig. 9 According to Peter Hamburger, there are 11 distinct, reducible, simple Venn s Diagrams. There are 8 distinct simple, irreducible Venn classes. Five of them have one convex drawing in the plane (and four of these were discovered by Grünbaum ). All five can be drawn with five congruent ellipses. Thus, in total, there are19 simple Venn classes for n = 5. Fig. 11 Peter Hamburger and Grünbaum are very important researchers about Venn s Diagrams. Fig. 12 3

4 C. Maximum number of sets intricate, that it is difficult to show in a single figure. Referring to the case n=7, Grünbaum [2] wrote: "at present it seems likely that no such diagram exists." However, Grünbaum found examples of such diagrams [3] and in 1992 additional examples were also discovered by Anthony Edwards and independently by Carla Savage and Peter Winkler. One of Grünbaum's examples is a remarkable non-monotone symmetric Venn diagram. There are 17 simple monotone symmetric Venn diagrams for n=7. Below we summarize what is know about symmetric Venn diagrams for n=7. D. Conclusions During the elaboration of this work I could see that exist a lot of form which I could represent Venn s Diagrams. About the question present at begin of this paper doesn t have an exact answer, but I tried show the results of researches that there are today. Fig. 13 Recently, Peter Hamburger [4] has constructed a symmetric Venn diagram for n=11. The diagram is monotone and highly non-simple. It is very similar in form to the diagram for n=7, but is vastly more complicated because of increase in the number of regions, intersection points, etc. In fact, the diagram is so E. References [1] J. Venn, On the diagrammatic and mechanical representation of propositions and reasonings, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 9 (1880) [2] Branko Grünbaum, Venn diagrams and Independent Families of Sets, Mathematics Magazine, 48 4

5 (Jan-Feb 1975) [Grünbaum awarded the MAA Lester R. Ford prize for this paper in 1976 (see AMM, Aug-Sept. 1976, pg. 587).] [3] Branko Grünbaum, Venn Diagrams II, Geombinatorics, Volume II, Issue 2, (1992) [4] Peter Hamburger, Doodles and Doilies, manuscript, ) eys/ds5/vennejc.html 2) Venn diagrams eobrien/ venn4.html 5