Venn diagram
A Venn diagram is a widely recognized graphic organizer that visually represents the logical relationships between sets or groups of items through overlapping circles or curves. Originating from the work of British mathematician John Venn in the late 19th century, these diagrams enable users to compare and contrast various attributes, highlighting similarities and differences between the sets. The intersections of the circles indicate shared attributes, while areas exclusive to a circle represent unique characteristics of that set.
Venn diagrams can illustrate relationships among two or more sets, with the basic format being two or three overlapping circles. While they are effective for visualizing the connections among fewer sets, representing four or more sets becomes increasingly complex, often leading to the use of ellipses or alternative methods. Venn diagrams have applications across multiple disciplines, including mathematics, logic, and computer science, serving as a valuable tool for teaching higher-order thinking skills. Their enduring relevance demonstrates their effectiveness in helping users better understand logical relationships and set theory concepts.
Venn diagram
The Venn diagram is one of the world’s most recognizable graphic organizers. Merriam-Webster defines the Venn diagram as “a graph that employs closed curves and especially circles to represent logical relations between and operations on sets and the terms of propositions by the inclusion, exclusion, or intersection of the curves.” A Venn diagram enables one to compare and contrast information and examine relationships between and among sets or groups of items. The graphic representation bears the name of the British mathematician who synthesized it, John Venn.
Background
Some of the earliest writings on visual representations of logical principles come from Ramon Llull, a Majorcan philosopher and theologian (ca. 1232–ca. 1315). Llull’s work would greatly influence the German philosopher and mathematician Gottfried Wilhelm Leibniz (1646–1716), who applied this knowledge to the creation of mechanical calculating wheels and developing the binary number system. Leonhard Euler (1707–1783), a Swiss mathematician, illustrated the logic of the syllogism (i.e., “If A is like B, and B is like C, then A must be like C”) with circles and ovals. Most of his representations involved concentric circles, describing shared and unshared attributes, as well as subsets (represented by B, a circle within a circle):
FPO: Place image Neither‗A‗Nor‗B.jpg here
English logician and philosopher John Venn (1834–1923) first presented his Euler-inspired system of graphic representation in an 1880 article published in the Philosophical Magazine and Journal of Science called “On the Diagrammatic and Mechanical Representation of Propositions and Reasonings.” He refined these graphics to express comparisons and contrasts among sets of attributes. Even with the refinements he added, Venn always referred to his diagrams as “Eulerian circles.” It was American philosopher Clarence Irving Lewis (1883–1964) who coined the term Venn diagram in his 1918 work, A Survey of Symbolic Logic.
Scottish philosopher Sir William Hamilton (1788–1856)—a contemporary of Venn’s—credited the German pedagogue Christian Weise (1642–1708) with using graphic diagrams to show the logical relationship between and among sets; Venn disputed that assertion.
Additionally, the work of two prominent British mathematicians greatly influenced John Venn. In the same year, 1847, Augustus De Morgan (1806–1871) wrote Formal Logic, and George Boole (1815–1864) penned The Mathematical Analysis of Logic. It was the latter who set down the distributive property of multiplication. Continuing with his 1854 treatise An Investigation of the Laws of Thought, Boole’s approach to mathematical logic would greatly inspire Venn, along with today’s computer scientists.
Overview
Venn diagrams, in their basic form, serve to illustrate comparisons and contrasts of attributes among two or more sets (items) or concepts. Where the ovals or circles intersect, the attributes in those segments are alike. Attributes in an oval or a circle not in a common area are specific to that set (item) or concept. The first example below compares and contrasts the sets A and B. The attributes in the area where the circles intersect are common to both A and B. The area outside both circles may contain attributes common to neither A nor B.
FPO: Place image Neither‗A‗Nor‗B‗2.jpg here
Two sets have two circles; three sets have three circles: A, B, and C. Attributes common to two sets are shown in three spaces, while attributes common to all three are shown in the middle.
FPO: Place image Not‗A‗Or‗B‗Or‗C‗3.jpg here
Four circles do not work, as there are not enough areas that overlap. For four sets, Venn used ellipses. However, in Symbolic Logic, he acknowledges the unwieldy nature of diagrams involving more than four terms.
Venn diagrams continue to be used to teach higher-order thinking skills in mathematics at all levels. Moreover, Venn diagrams serve to illustrate logical relationships among sets or attributes in many fields to this day.
Bibliography
Alsina, Claudi, and Roger Nelsen. Icons of Mathematics: An Exploration of Twenty Key Images. Washington, DC: Mathematical Assn. of Amer., 2011. Print.
Barnett, Janet Heine. “Origins of Boolean Algebra in the Logic of Classes: George Boole, John Venn and C. S. Peirce.” MAA.org. Mathematical Assn. of Amer., July 2013. Web. 3 Sept. 2014.
Baron, M. E. “A Note on the Historical Development of Logic Diagrams.” Mathematical Gazette 53.384(1969): 113–25. Print.
Boswell, Kelly. Diagrams, Diagrams, Diagrams! North Mankato: Capstone, 2014. Print.
Edwards, A. W. F. Cogwheels of the Mind: The Story of Venn Diagrams. Baltimore: Johns Hopkins P, 2004. Print.
Pinter, Charles C. A Book of Set Theory. Mineola: Dover, 2013. Print.
Ruskey, Frank, Carla D. Savage, and Stan Wagon. “The Search for Simple Symmetric Venn Diagrams.” Notices of the AMS 53.11 (2006): 1304–11. Print.
Ruskey, Frank, and Mark Weston. “A Survey of Venn Diagrams.” Combinatorics.org. Electronic Jour. of Combinatorics, 18 June 2005. Web. 3 Sept. 2014.