Euler diagram

An Euler diagram (/ˈɔɪlər/, OY-lər) is a diagrammatic means of representing sets and their relationships. Typically they involve overlapping shapes, and may be scaled, such that the area of the shape is proportional to the number of elements it contains. They are particularly useful for explaining complex hierarchies and overlapping definitions. They are often confused with Venn diagrams. Unlike Venn diagrams, which show all possible relations between different sets, the Euler diagram shows only relevant relationships.Euler diagramVenn diagramA Venn diagram shows all possible intersections.An Euler diagram illustrating that the set of 'animals with four legs' is a subset of 'animals', but the set of 'minerals' is disjoint (has no members in common) with 'animals'Euler diagram visualizing a real situation, the relationships between various supranational European organizations. (clickable version)Humorous diagram comparing Euler and Venn diagrams.Euler diagram of types of triangles, using the definition that isosceles triangles have at least (rather than exactly) 2 equal sides.Euler diagram of terminology of the British Isles.The 22 (of 256) essentially different Venn diagrams with 3 circles (top) and their corresponding Euler diagrams (bottom)Some of the Euler diagrams are not typical, and some are even equivalent to Venn diagrams. Areas are shaded to indicate that they contain no elements. An Euler diagram (/ˈɔɪlər/, OY-lər) is a diagrammatic means of representing sets and their relationships. Typically they involve overlapping shapes, and may be scaled, such that the area of the shape is proportional to the number of elements it contains. They are particularly useful for explaining complex hierarchies and overlapping definitions. They are often confused with Venn diagrams. Unlike Venn diagrams, which show all possible relations between different sets, the Euler diagram shows only relevant relationships. The first use of 'Eulerian circles' is commonly attributed to Swiss mathematician Leonhard Euler (1707–1783). In the United States, both Venn and Euler diagrams were incorporated as part of instruction in set theory as part of the new math movement of the 1960s. Since then, they have also been adopted by other curriculum fields such as reading as well as organizations and businesses. Euler diagrams consist of simple closed shapes in a two dimensional plane that each depict a set or category. How or if these shapes overlap demonstrates the relationships between the sets. There are only 3 possible relationships between any 2 sets; completely inclusive, partially inclusive, and exclusive. This is also referred to as containment, overlap or neither or, especially in mathematics, it may be referred to as subset, intersection and disjoint. Each Euler curve divides the plane into two regions or 'zones': the interior, which symbolically represents the elements of the set, and the exterior, which represents all elements that are not members of the set. Curves whose interior zones do not intersect represent disjoint sets. Two curves whose interior zones intersect represent sets that have common elements; the zone inside both curves represents the set of elements common to both sets (the intersection of the sets). A curve that is contained completely within the interior zone of another represents a subset of it. Venn diagrams are a more restrictive form of Euler diagrams. A Venn diagram must contain all 2n logically possible zones of overlap between its n curves, representing all combinations of inclusion/exclusion of its constituent sets. Regions not part of the set are indicated by coloring them black, in contrast to Euler diagrams, where membership in the set is indicated by overlap as well as color. When the number of sets grows beyond 3 a Venn diagram becomes visually complex, especially compared to the corresponding Euler diagram. The difference between Euler and Venn diagrams can be seen in the following example. Take the three sets: