“The interior regions created by intersections in a Venn diagram do not need to be circular.”

Polygons can draw accurate areaproportional Venn diagrams for any data with three sets [30] , but as shown in Figure 1D -F, their non-smooth and nonsymmetric curves are not easily distinguishable and impede comprehension [31, 32] . Despite these problems, current drawing methods use either circles or polygons.… In some cases, the generated diagrams do not depict all the required overlaps between the curves, as demonstrated in Section 4.3 and Section 4. 4 . Such area-proportional Venn diagrams cannot be drawn analytically using a specific curve shape and so numerical methods or heuristics are required [30] . Circles can draw Venn diagrams with region areas that are proportional to any data with two sets [5] , but not three [30] due to their limited degrees of freedom (i.e., a centre and a radius). Polygons can draw accurate areaproportional Venn diagrams for any data with three sets [30] , but as shown in Figure 1D -F, their non-smooth and nonsymmetric curves are not easily distinguishable and impede comprehension [31, 32] . Despite these problems, current drawing methods use either circles or polygons. Ellipses have more degrees of freedom (i.e., a centre, two semiaxes, an angle of rotation) than circles and are similarly smooth. So diagrams drawn with ellipses are more likely to be accurate with respective to the required quantitative data and easy to comprehend due to their distinguishable curves. This is illustrated in Figure 2 where the diagrams accurately depict the quantities indicated by the numeric labels of the respective diagram in Figure 1 . …

Some other constructions exist, using ellipses or other curves with no particular shape, but they have the same flaw: they are not really easy to read and to draw on the blackboard.Introduction As teachers, we use Venn diagrams with three curves to teach our students bases of the set theory. (Un)fortunately, because some students are more curious than others, we were asked to draw Venn diagrams with four or five sets. Here is now the problem: such diagrams cannot be drawn using circles. Some other constructions exist, using ellipses or other curves with no particular shape, but they have the same flaw: they are not really easy to read and to draw on the blackboard. The regions do not all have the same size and it can be hard to determine whether a region is included in a given set or not. We became interested in the construction of an easy-to-read Venn diagram with more than four sets. The Carrol-Lewis diagram [3] persuades us because of its simplicity and its ease to be drawn on a blackboard. We present in this paper a construction which extends it to any number of sets.

Fortunately, it is known that far more accurate diagrams can be drawn when ellipses are used in place of circles, with accuracy rates well over 90% achievable for Venn-3 diagrams [3] .Until recently, users had a choice between reasonably accurate ellipse-based diagrams limited to three sets [3] , or the typically inaccurate diagrams drawn with circle-based methods [4] , which allow a larger number of sets. Tools that use 3 circles forming Venn-3 are known to be inaccurate [5] . Inaccuracies worsen as the number of sets increases. Fortunately, it is known that far more accurate diagrams can be drawn when ellipses are used in place of circles, with accuracy rates well over 90% achievable for Venn-3 diagrams [3] . Prior to the systems discussed in this paper, no method has been developed to draw diagrams beyond three sets with ellipses. The earliest area-proportional work considered Venn-2 drawn as two circles [6] , which can always be drawn exactly no matter the cardinality of the set intersections. However the subsequent work on circle-based Venn-3 diagrams [7, 8] , demonstrated the inaccuracy inherent in this more complex type of diagram. …

Let C = { C 1, C 2, ..., C n } be a collection of simple closed curves drawn in the plane. … The figure below is a Venn diagram of 4 ellipses, originally found by Venn himself [Ve80].What is a Venn Diagram? We follow Grünbaum [Gr75] in first defining a more general concept, an independent family. Let C = { C 1, C 2, ..., C n } be a collection of simple closed curves drawn in the plane. The collection C is said to be an independent family if the region formed by the intersection of X 1, X 2, ..., X n is nonempty, where each X i is either int(C i ) (the interior of C i ) or is ext(C i ) (the exterior of C i ). If, in addition, each such region is connected and there are only finitely many points of intersection between curves, then C is a Venn diagram, or an n-Venn diagram if we wish to emphasize the number of curves in the diagram. … On the left we show an example, for n=4, of an independent family that is not a Venn diagram. Here the simple closed curves are each congruent triangles, which are colored red, blue, green, and gold. Note that the intersection of the interiors of the red and blue triangles with the exteriors of the green and gold triangles gives a disconnected region, the shaded region(s) in the figure. The figure below is a Venn diagram of 4 ellipses, originally found by Venn himself [Ve80]. See also a black-and-white version, and its Tutte embedding 11 (these last two figures provided by Stuart Anderson).

Let C = { C 1, C 2, ..., C n } be a collection of simple closed curves drawn in the plane. … The figure below is a Venn diagram of 4 ellipses, originally found by Venn himself [Ve80].We follow Grünbaum [Gr75] in first defining a more general concept, an independent family. Let C = { C 1, C 2, ..., C n } be a collection of simple closed curves drawn in the plane. The collection C is said to be an independent family if the region formed by the intersection of X 1, X 2, ..., X n is nonempty, where each X i is either int(C i ) (the interior of C i ) or is ext(C i ) (the exterior of C i ). If, in addition, each such region is connected and there are only finitely many points of intersection between curves, then C is a Venn diagram, or an n-Venn diagram if we wish to emphasize the number of curves in the diagram. … On the left we show an example, for n=4, of an independent family that is not a Venn diagram. Here the simple closed curves are each congruent triangles, which are colored red, blue, green, and gold. Note that the intersection of the interiors of the red and blue triangles with the exteriors of the green and gold triangles gives a disconnected region, the shaded region(s) in the figure. The figure below is a Venn diagram of 4 ellipses, originally found by Venn himself [Ve80]. See also a black-and-white version, and its Tutte embedding 11 (these last two figures provided by Stuart Anderson).

Furthermore, some approaches to layout attempt to find area proportional diagrams with unique labelling and the property that each curve is a circle [CR05a, KMGB05] . Exact area proportional layouts under these conditions is not always possible.… The weight function does not assign an area to the zone which is not inside any curves. In figure 7 There are obvious variations of this property; for example, when drawing bounding rectangles around Euler diagrams, rather than embedding them in the whole of R 2 , one can stipulate the area of all of the zones. The work on laying out area proportional Euler diagrams typically produces diagrams that have connected zones [CR03, CR05b, KMGB05] . Furthermore, some approaches to layout attempt to find area proportional diagrams with unique labelling and the property that each curve is a circle [CR05a, KMGB05] . Exact area proportional layouts under these conditions is not always possible. The desirability of utilizing circles and getting an approximate result has also led to the notion of relative size, so that more diagrams can be drawn where the size of one zone is specified to be bigger than another [CR05a] . Connected Diagram Property

Let C = { C 1,..., C n} be a collection of simple closed curves drawn in the plane. The collection C is said to be an independent family if the intersection of X 1, X 2, ..., X n is nonempty, where each X i is either int(C i) (the interior of C i) or is ext(C i) (the exterior of C i). If, in addition, each such intersection is connected, then C is a Venn diagram, or an n-Venn diagram if we wish to emphasize the number of curves in the diagram.What is a Venn Diagram? We follow Grünbaum [Gr75] in first defining a more general concept, an independent family. Let C = { C 1,..., C n} be a collection of simple closed curves drawn in the plane. The collection C is said to be an independent family if the intersection of X 1, X 2, ..., X n is nonempty, where each X i is either int(C i) (the interior of C i) or is ext(C i) (the exterior of C i). If, in addition, each such intersection is connected, then C is a Venn diagram, or an n-Venn diagram if we wish to emphasize the number of curves in the diagram. On the left we show the most familiar of all Venn diagrams. In this case n=3, the simple closed curves are all circles, and in the leftmost diagram the 8 regions have been labelled with the interiors that are included in each intersection. The eighth region is the outside, corresponding to the empty set. The colored diagram has all 1-sets colored yellow, all 2-sets red, and the 3-set blue. This diagram also occurs as a minimal projection of the Borromean Rings.

Euler always drew his diagrams as collections of circles, but remarks in Letter CIII, These circles, or rather these spaces, for it is of no importance of what figure they are of,....LETTER CII. Of the Perfections of a Language. Judgements and Nature of Propositions, affirmative and negative; universal or particular. LETTER CIII. Of Syllogisms, and their different Forms, when the first Proposition is universal. LETTER CIV. Different Forms of Syllogisms, whose first Proposition is particular. LETTER CV. Analysis of some Syllogisms. Euler always drew his diagrams as collections of circles, but remarks in Letter CIII, These circles, or rather these spaces, for it is of no importance of what figure they are of,.... We know of no written record that Euler ever made use of the familiar 3 circle Venn diagram. Chapter V of John Venn's book Symbolic Logic contains an explanation and comparison of Venn and Euler diagrams [Ve81]. The second part of his historical notes contains much information about the history of diagrammic reasoning.

Venn constructed diagrams of up to five simply connected regions that overlapped each other once in each possible way of overlapping. … “But for merely theoretical purposes the rule of formulation would be very simple. It would merely be to begin by drawing any closed figure, and then proceed to draw others, subject to the one condition that each is to intersect once and once only all the existing subdivisions produced by those which had gone before.”* Article * Metrics Article contents * Extract * Footnotes * References Get access Share CiteRights & Permissions[Opens in a new window] Extract Venn constructed diagrams of up to five simply connected regions that overlapped each other once in each possible way of overlapping. Although Venn did not prove that his diagrams were constructible for more than five simply connected regions —in fact, he preferred to have a doubly connected region in his 5-class diagram —he summarized his method of construction with an intuitive argument: “But for merely theoretical purposes the rule of formulation would be very simple. It would merely be to begin by drawing any closed figure, and then proceed to draw others, subject to the one condition that each is to intersect once and once only all the existing subdivisions produced by those which had gone before.” The method of construction given below leads to a simple topological proof that Venn diagrams can be constructed for any number of simply connected regions. TypeResearch Article Information The Journal of Symbolic Logic,Volume 24,Issue 4, December 1959, pp. 303 - 304 DOI:https://doi.org/10.2307/2963899[Opens in a new window] Copyright Copyright ©Association for Symbolic Logic 1952 Access options Get access to the full version of this content by using one of the access options below. …

In general, an order- Venn diagram is a collection of simple closed curves in the plane such that 1. The curves partition the plane into connected regions, and 2. Each subset of corresponds to a unique region formed by the intersection of the interiors of the curves in (Ruskey).The order-three diagram (right) consists of three symmetrically placed mutually intersecting circles comprising a total of eight regions. The regions labeled , , and consist of members which are only in one set and no others, the three regions labelled , , and consist of members which are in two sets but not the third, the region consists of members which are simultaneously in all three, and no regions occupied represents . In general, an order- Venn diagram is a collection of simple closed curves in the plane such that 1. The curves partition the plane into connected regions, and 2. Each subset of corresponds to a unique region formed by the intersection of the interiors of the curves in (Ruskey). Since there are (the binomial coefficient) ways to pick members from a total of , the number of regions in an order Venn diagram is

A Venn diagram of $n$ variables $a_1,\dotsc,a_n$ of classical propositional logic is a selection of closed contours $C_1,\dotsc,C_n$ (with homeomorphic circumferences) which subdivides the plane into $2^n$ domains, some of which (e.g. $v_1,\dotsc,v_k$, $0\leq k\leq2^n$) are marked.Venn diagram A graphic representation of formulas of mathematical logic, mainly formulas of the propositional calculus. A Venn diagram of $n$ variables $a_1,\dotsc,a_n$ of classical propositional logic is a selection of closed contours $C_1,\dotsc,C_n$ (with homeomorphic circumferences) which subdivides the plane into $2^n$ domains, some of which (e.g. $v_1,\dotsc,v_k$, $0\leq k\leq2^n$) are marked. Each marked domain $v_i$, $0<i\leq k$, is put into correspondence with the formula $B_i=b_1\mathbin{\&}\dotsb\mathbin{\&}b_n$ where $b_j$, $0<j\leq n$, is $a_j$ if $v_i$ lies within the contour $C_j$ and $b_j$ is $\neg a_j$ otherwise. The formula corresponding to the diagram as a whole is $B_1\lor\dotsb\lor B_n$. Thus, the Venn diagram in the figure corresponds to the formula

Alternatives for use in logic include Veitch diagrams and Karnaugh maps [13], however these take a rectilinear approach to visualizing intersection, which is not always effective from a usability perspective.… There is potential for confusion if the user interprets an ordering to the line that is not present or if a line follows a path with many bends. Hypergraphs can also be used for grouping items if drawn in the subset standard [8], however, this method can lead to unwanted, empty, set intersections being present and relies on the notion that items are already laid out, problems that also occur with Bubble Sets [27]. Alternatives for use in logic include Veitch diagrams and Karnaugh maps [13], however these take a rectilinear approach to visualizing intersection, which is not always effective from a usability perspective. Area-proportional Euler diagrams aim to ensure that the regions are of a desired area. This is similar to cartograms [30], where territories are distorted so that their area is of a desired value, representing a quantity associated with the territory (for example, population). …

What counts as a wellformedness property is not easily defined, and many other features of diagrams can also be legitimately called wellformedness properties. One such is that of restricting the shape of curves in some way, for example, a property could be defined that ensured that all curves were circles (or more generally ellipses), or that all curves were rectilinear.… See Figure 15 for examples of each: 1. n-points three or more curves cross at the same point (often called ‘triple points’, especially when three curves meet). 2. concurrency two or more curve segments are con current. 3. duplicate curve labels two or more curves have the same label. 4. non-simple curves a curve self intersects. 5. disconnected zone a zone consists of more than one minimal region. 6. brushing points two or more curves meet at a point but do not cross. What counts as a wellformedness property is not easily defined, and many other features of diagrams can also be legitimately called wellformedness properties. One such is that of restricting the shape of curves in some way, for example, a property could be defined that ensured that all curves were circles (or more generally ellipses), or that all curves were rectilinear. Another is that the zone areas R a) ∅ P Q R PQ PR QR with an n-point (n=3) P Q R

A Venn diagram is a visual representation using closed curves, typically circles, to illustrate the relationships between sets or classes.What is a Venn diagram? A Venn diagram is a visual representation using closed curves, typically circles, to illustrate the relationships between sets or classes. These diagrams, introduced by English logician John Venn, depict logical relations of inclusion and exclusion. Venn diagrams are used to test the validity of categorical syllogisms. They employ lowercase x’s and shading to indicate the existence and nonexistence, respectively, of members within a class. Venn diagrams have been a standard part of introductory logic since the mid-20th century. 3 Britannica Sources

The region of intersection of three symmetrically placed circles (as in a Venn diagram), in the special case of the center of each being located at the intersection of the other two, is called a Reuleaux triangle.The circle is a conic section obtained by the intersection of a cone with a plane perpendicular to the cone's symmetry axis. It is also a Lissajous curve. A circle is the degenerate case of an ellipse with equal semimajor and semiminor axes (i.e., with eccentricity 0). The interior of a circle is called a disk. The generalization of a circle to three dimensions is called a sphere, and to dimensions for a hypersphere. The region of intersection of two circles is called a lens. The region of intersection of three symmetrically placed circles (as in a Venn diagram), in the special case of the center of each being located at the intersection of the other two, is called a Reuleaux triangle. In Cartesian coordinates, the equation of a circle of radius centered on is

In this illustration, additional zones were inserted so that the shapes can be circles. This, however, is not misleading, because the elements are also drawn inside their respective zones, indicating that some faces are not zones.C A ⊃ C C ⊃ C B . The layout of a DAG can also be used to visualize the clustering (Figure 3 .1), but we will use this constructed DAG only as a supporting data structure later on. The duality of DAG and Euler diagrams is illustrated in Figure 3 .1. Figure 3 .1: A clustering graph and its corresponding Euler representation. In this illustration, additional zones were inserted so that the shapes can be circles. This, however, is not misleading, because the elements are also drawn inside their respective zones, indicating that some faces are not zones. As a hierarchy can be represented by a rooted tree, an arbitrary clustering may be represented by a directed acyclic graph G = (V(G), E(G)). The vertex set V(G) is then simply one vertex for each element and cluster. The edges E(G) correspond to the clustering structure, i.e. there is a directed edge from u to v, if and only if v is an element and u is a cluster that contains this element or if v is a subcluster of u. The sinks of G, i.e. …

A Venn diagram uses simple closed curves on a plane to represent sets. The curves are often circles or ellipses. … Shapes other than circles can be employed as shown below by Venn's own higher set diagrams.Venn diagram showing the uppercase glyphs shared by the Greek (upper left), Latin (upper right), and Russian Cyrillic (bottom) alphabets A Venn diagram is a widely used diagram style that shows the logical relation between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships in probability, logic, statistics, linguistics and computer science. A Venn diagram uses simple closed curves on a plane to represent sets. The curves are often circles or ellipses. Very similar ideas had been proposed before Venn such as by Christian Weise in 1712 (Nucleus Logicoe Wiesianoe) and Leonhard Euler in 1768 (Letters to a German Princess). The idea was popularised by Venn in Symbolic Logic, Chapter V "Diagrammatic Representation", published in 1881. … [18]: 157 Venn diagrams normally comprise overlapping circles. The interior of the circle symbolically represents the elements of the set, while the exterior represents elements that are not members of the set. For instance, in a two-set Venn diagram, one circle may represent the group of all wooden objects, while the other circle may represent the set of all tables. The overlapping region, or intersection, would then represent the set of all wooden tables. Shapes other than circles can be employed as shown below by Venn's own higher set diagrams. Venn diagrams do not generally contain information on the relative or absolute sizes (cardinality) of sets. That is, they are schematic diagrams generally not drawn to scale. Venn diagrams are similar to Euler diagrams. However, a Venn diagram for n component sets must contain all 2 n hypothetically possible zones, that correspond to some combination of inclusion or exclusion in each of the component sets. [28] Euler diagrams contain only the actually possible zones in a given context. …

Figure 4 shows examples of Venn diagrams drawn using ellipses [11] and triangles [3].An interesting problem popularized by Gr¨unbaum [11, 12, 13, 14] is to consider which Venn diagrams can be drawn using specific shapes. Figure 2 shows a 3-Venn diagram comprised of circles; a natural question to ask is if such a diagram exists for four sets. It turns out the answer is no, and this can be proved easily using Euler’s formula (V = E − F + 2) and noting that such a diagram must have 2 4 = 16 faces and that two circles can intersect at at most two points [22]. Figure 4 shows examples of Venn diagrams drawn using ellipses [11] and triangles [3]. The diagram in Fig. 4(a) is special because it is an example of a symmetric Venn diagram; that is, a diagram with n-fold rotational symmetry and (necessarily) congruent curves. Symmetric Venn diagrams exist if and only if n is prime [10]. On his “Math Recreations” web site [24], Mark Thompson proposed the novel problem of finding Venn polyominoes (from now on referred to as n polyVenns); these are Venn diagrams whose curves are the outlines of poly ominoes. …

Furthermore, some approaches to layout at tempt to find area proportional diagrams with unique labelling and the property that each curve is a circle [CR05a, KMGB05]. Exact area proportional layouts under these conditions is not always possible.be a function. The diagram d possesses the w-zone area property if and only if all of the zones, z in the domain of w have area w(z). There are obvious variations of this property; for example, when drawing bounding rectangles around Euler diagrams, rather than embedding them in the whole of R 2, one can stipulate the area of all of the zones. The work on laying out area proportional Euler diagrams typically produces diagrams that have connected zones [CR03, CR05b, KMGB05]. Furthermore, some approaches to layout at tempt to find area proportional diagrams with unique labelling and the property that each curve is a circle [CR05a, KMGB05]. Exact area proportional layouts under these conditions is not always possible. The desirability of utilizing circles and getting an approximate result has also led to the notion of relative size, so that more diagrams can be drawn where the size of one zone is specified to be bigger than another [CR05a]. 3.5 Connected Diagram Property

The algorithms are presented for generic curves, allowing for specialisations such as utilising fixed geometric shapes for curves that often occur in applications. … Placing restrictions on the geometric shapes used for contours (which is common in some applications) can enable particularly fast computations. For example, if each contour is a simple geometric shape, such as a circle or an ellipse, these computations reduce to solving a system of two quadratic equations… We present a variation and extension of the methodology which enables region computations for Euler diagrams under the relaxation of several drawing conventions. We provide complexity analysis and compare with the previous methodology. The algorithms are presented for generic curves, allowing for specialisations such as utilising fixed geometric shapes for curves that often occur in applications. 1 Introduction … Fig. 8. The addition of contour A splits eight minimal regions determined by the eight arcs that comprise A. However, it splits nine zones, eight of which are the distinct zones containing the eight minimal regions that are split. The ninth zone { B} is split, without splitting any of its constituent minimal regions, since one of its minimal regions is covered by A but the other is not. Placing restrictions on the geometric shapes used for contours (which is common in some applications) can enable particularly fast computations. For example, if each contour is a simple geometric shape, such as a circle or an ellipse, these computations reduce to solving a system of two quadratic equations (1 and 2) and a quadratic equa tion (3), which can be computed very quickly (with different methods having different time/precision tradeoffs). Algorithm ComputeContourRelationships: (i) computes the relationship between the contours present in a diagram d (with WF3 relaxed) and a contour A; (ii) updates the set of crossing points of d. We will refer to Fig. 8 to assist with the explanation of the algorithms. Consider the addition of the dashed contour A to the diagram in Fig. …

All Venn diagram descriptions are drawable with simple closed curves and connected minimal regions [25] .… An element of W is the set of labels of the curves that contain the corresponding minimal region. For example, (W = {{A}, {B}, {A, B}, {A, B, C}}, L = {A, B, C}) describes the Euler diagram in figure 1 . The minimal region which is inside A but outside B and C corresponds to the set {A} in W . If W = PL -{∅} then (W, L) describes a Venn diagram. All Venn diagram descriptions are drawable with simple closed curves and connected minimal regions [25] . Simple Closed Curves

A family C = {C1, C2,...,Cn} of simple (Jordan) curves in the plane is a Venn diagram provided each of the 2n sets X1 ↔ X2 ↔ ... ↔ Xn, where each Xj is one of the two connected components of the complement of Cj (that is, each Xj is either the bounded interior or the unbounded exterior of Cj ), is nonempty and connected.Almost every book on discrete mathematics shows some or all of the Venn diagrams of one, two, or three circles, shown in Figure 1. Named after the logician John Venn, these diagrams are meant to help in discussing de tails of logical possibilities. Modifying work by various predecessors (including Euler), Venn proposed in [6] and popularized in [7] a definition which can be formu lated as follows. A family C = {C1, C2,...,Cn} of simple (Jordan) curves in the plane is a Venn diagram provided each of the 2n sets X1 ↔ X2 ↔ ... ↔ Xn, where each Xj is one of the two connected components of the complement of Cj (that is, each Xj is either the bounded interior or the unbounded exterior of Cj ), is nonempty and connected. The last two attributes are the determining ones. Clearly, the families of circles in Figure 1 are Venn diagrams, while the families of curves in Figure 2 are not. It is to be regretted that no book on discrete mathematics pursues the topic much further –– it leads to fascinating mathematics of an elementary character. In this note and in a sequel we shall discuss various results and open questions concerning Venn diagrams.

Whilst circles are widely used to draw such diagrams, most area specifications cannot be drawn in this way and, so, should only be used where an approximate solution is acceptable. … In this paper, we explore the use of convex shapes for drawing exact area proportional Venn-3 diagrams. … We describe methods for constructing convex diagrams with polygons that have four or five sides and derive results concerning which area specifications can be drawn with them.… In these diagrams, the areas of the regions are in proportion to the given values. Venn-3, the Venn diagram consisting of three intersecting curves, has been used in many applications, including marketing, ecology and medicine. Whilst circles are widely used to draw such diagrams, most area specifications cannot be drawn in this way and, so, should only be used where an approximate solution is acceptable. However, placing different restrictions on the shape of curves may result in usable diagrams that have an exact solution, that is, where the areas of the regions are exactly in proportion to the represented data. In this paper, we explore the use of convex shapes for drawing exact area proportional Venn-3 diagrams. Convex curves reduce the visual complexity of the diagram and, as most desirable shapes (such as circles, ovals and rectangles) are convex, the work described here may lead to further drawing methods with these shapes. We describe methods for constructing convex diagrams with polygons that have four or five sides and derive results concerning which area specifications can be drawn with them. This work improves the state-of-the-art by extending the set of area specifications that can be drawn in a convex manner. We also show how, when a specification cannot be drawn in a convex manner, a non-convex drawing can be generated. ...Read more Related papers …

Each set or subset is represented by a circle or a blob of some other shape, as shown in the diagram.Encyclopædia Britannica, Inc. An excellent way of representing subsets and the like is the use of Venn diagrams. Each set or subset is represented by a circle or a blob of some other shape, as shown in the diagram. One circle shown inside another means that one set is contained in the other, as the set of acute triangles is shown within the set of oblique triangles. The area where two circles overlap represents the intersection of sets, as that for acute-isosceles triangles, represented by the purple region. Every kind of triangle fits into this diagram somewhere. It is, in fact, the same diagram as the previous one, as can be seen by comparing the colors. …

Each set in a Venn diagram is depicted by drawing a simple closed curve — typically a circle, but not necessarily.Figure 4.2: A prototypical Venn digram. In a Venn diagram, the universe of discourse is normally drawn as a rectangular region inside of which all the action occurs. Each set in a Venn diagram is depicted by drawing a simple closed curve — typically a circle, but not necessarily. For instance, if you want to draw a Venn diagram that shows all the possible intersections among four sets, you’ll find it’s impossible with (only) circles. Figure 4.3 shows a drawing of a Venn diagram for four sets. Figure 4.3: A four-set Venn diagram.

A Venn diagram represents certain sets of polygons, with each region in the form of the polygon it represents.Cite (Subscriber Feature) A Venn diagram represents certain sets of polygons, with each region in the form of the polygon it represents. © Encyclopædia Britannica, Inc.

Euler diagrams consist of simple closed shapes in a two-dimensional plane that each depict a set or category.… Euler diagrams were first developed for logic, especially syllogistics, and only later transferred to set theory. 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 whether these shapes overlap demonstrates the relationships between the sets. Each 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 which do not overlap represent disjoint sets, which have no elements in common. …

There are examples of Euler diagrams with 9 sets that are not drawable using simple closed curves without the creation of unwanted zones since they would have to have non-planar dual graphs.… For example, connectedness of zones might be enforced, or concurrency of curves or multiple points might be banned, as might tangential intersection of curves. In the adjacent diagram, examples of small Venn diagrams are transformed into Euler diagrams by sequences of transformations; some of the intermediate diagrams have concurrency of curves. However, this sort of transformation of a Venn diagram with shading into an Euler diagram without shading is not always possible. There are examples of Euler diagrams with 9 sets that are not drawable using simple closed curves without the creation of unwanted zones since they would have to have non-planar dual graphs. Example: Euler- to Venn-diagram and Karnaugh map

It consists of overlapping circles or closed curves, where each circle represents a set, and the overlapping regions represent the intersection of those sets.What is a Venn Diagram? A Venn diagram is a graphical representation of sets and their relationships, particularly their intersections and unions. It consists of overlapping circles or closed curves, where each circle represents a set, and the overlapping regions represent the intersection of those sets. Venn diagrams are named after the British logician John Venn, who popularized their use in the late 19th century. What is Euler Diagram?