## Venn diagram

Diagram that shows all possible logical relations between a collection of sets

A **Venn diagram** is a widely used diagram style that shows the logical relation between sets, popularized by John Venn 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 drawn on a plane to represent sets. Very often, these curves are circles or ellipses.

Similar ideas had been proposed before Venn. Christian Weise in 1712 (*Nucleus Logicoe Wiesianoe*) and Leonhard Euler (*Letters to a German Princess*) in 1768, for instance, came up with similar ideas. The idea was popularised by Venn in *Symbolic Logic*, Chapter V "Diagrammatic Representation", 1881.

### Details[edit]

A Venn diagram may also be called a **primary diagram**, **set diagram** or **logic diagram**. It is a diagram that shows *all* possible logical relations between a finite collection of different sets. These diagrams depict elements as points in the plane, and sets as regions inside closed curves. A Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set. The points inside a curve labelled *S* represent elements of the set *S*, while points outside the boundary represent elements not in the set *S*. This lends itself to intuitive visualizations; for example, the set of all elements that are members of both sets *S* and *T*, denoted *S* ∩ *T* and read "the intersection of *S* and *T*", is represented visually by the area of overlap of the regions *S* and *T*.^{[1]}^{[2]}

In Venn diagrams, the curves are overlapped in every possible way, showing all possible relations between the sets. They are thus a special case of Euler diagrams, which do not necessarily show all relations. Venn diagrams were conceived around 1880 by John Venn. They are used to teach elementary set theory, as well as illustrate simple set relationships in probability, logic, statistics, linguistics, and computer science.

A Venn diagram in which the area of each shape is proportional to the number of elements it contains is called an **area-proportional** (or **scaled**) **Venn diagram**.

Venn diagrams are used heavily in the logic of class branch of reasoning.

### Example[edit]

This example involves two sets, A and B, represented here as coloured circles. The orange circle, set A, represents all types of living creatures that are two-legged. The blue circle, set B, represents the living creatures that can fly. Each separate type of creature can be imagined as a point somewhere in the diagram. Living creatures that can fly *and* have two legs—for example, parrots—are then in both sets, so they correspond to points in the region where the blue and orange circles overlap. This overlapping region would only contain those elements (in this example, creatures) that are members of both set A (two-legged creatures) and set B (flying creatures).

Humans and penguins are bipedal, and so are in the orange circle, but since they cannot fly, they appear in the left part of the orange circle, where it does not overlap with the blue circle. Mosquitoes can fly, but have six, not two, legs, so the point for mosquitoes is in the part of the blue circle that does not overlap with the orange one. Creatures that are not two-legged and cannot fly (for example, whales and spiders) would all be represented by points outside both circles.

The combined region of sets A and B is called the *union* of A and B, denoted by A ∪ B.^{[1]}^{[3]} The union in this case contains all living creatures that either are two-legged or can fly (or both).

The region included in both A and B, where the two sets overlap, is called the *intersection* of A and B, denoted by A ∩ B.^{[1]}^{[3]} In this example, the intersection of the two sets is not empty, because there *are* points that represent creatures that are in *both* the orange and blue circles.

### History[edit]

Venn diagrams were introduced in 1880 by John Venn in a paper entitled "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings^{[4]}" in the *Philosophical Magazine and Journal of Science*,^{[5]} about the different ways to represent propositions by diagrams.^{[6]}^{[7]}^{[8]} The use of these types of diagrams in formal logic, according to Frank Ruskey and Mark Weston, is "not an easy history to trace, but it is certain that the diagrams that are popularly associated with Venn, in fact, originated much earlier. They are rightly associated with Venn, however, because he comprehensively surveyed and formalized their usage, and was the first to generalize them".^{[9]}

Venn himself did not use the term "Venn diagram" and referred to his invention as "Eulerian Circles".^{[8]} For example, in the opening sentence of his 1880 article Venn writes, "Schemes of diagrammatic representation have been so familiarly introduced into logical treatises during the last century or so, that many readers, even those who have made no professional study of logic, may be supposed to be acquainted with the general nature and object of such devices. Of these schemes one only, viz. that commonly called 'Eulerian circles,' has met with any general acceptance..."^{[6]}^{[7]}Lewis Carroll (Charles L. Dodgson) includes "Venn's Method of Diagrams" as well as "Euler's Method of Diagrams" in an "Appendix, Addressed to Teachers" of his book *Symbolic Logic* (4th edition published in 1896). The term "Venn diagram" was later used by Clarence Irving Lewis in 1918, in his book *A Survey of Symbolic Logic*.^{[9]}^{[10]}

Venn diagrams are very similar to Euler diagrams, which were invented by Leonhard Euler in the 18th century.^{[note 1]}^{[11]}^{[12]}Margaret Baron has noted that Leibniz (1646–1716) produced similar diagrams before Euler in the 17th century, but much of it was unpublished.^{[13]} She also observes even earlier Euler-like diagrams by Ramon Llull in the 13th Century.^{[14]}

In the 20th century, Venn diagrams were further developed. David Wilson Henderson showed, in 1963, that the existence of an *n*-Venn diagram with *n*-fold rotational symmetry implied that *n* was a prime number.^{[15]} He also showed that such symmetric Venn diagrams exist when *n* is five or seven. In 2002, Peter Hamburger found symmetric Venn diagrams for *n* = 11 and in 2003, Griggs, Killian, and Savage showed that symmetric Venn diagrams exist for all other primes. These combined results show that rotationally symmetric Venn diagrams exist, if and only if *n* is a prime number.^{[16]}

Venn diagrams and Euler diagrams were incorporated as part of instruction in set theory, as part of the new math movement in the 1960s. Since then, they have also been adopted in the curriculum of other fields such as reading.^{[17]}

### Overview[edit]

See also: Set (mathematics) § Basic operations

A Venn diagram is constructed with a collection of simple closed curves drawn in a plane. According to Lewis,^{[10]} the "principle of these diagrams is that classes [or *sets*] be represented by regions in such relation to one another that all the possible logical relations of these classes can be indicated in the same diagram. That is, the diagram initially leaves room for any possible relation of the classes, and the actual or given relation, can then be specified by indicating that some particular region is null or is not-null".^{[10]}^{: 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.^{[18]} Euler diagrams contain only the actually possible zones in a given context. In Venn diagrams, a shaded zone may represent an empty zone, whereas in an Euler diagram, the corresponding zone is missing from the diagram. For example, if one set represents *dairy products* and another *cheeses*, the Venn diagram contains a zone for cheeses that are not dairy products. Assuming that in the context *cheese* means some type of dairy product, the Euler diagram has the cheese zone entirely contained within the dairy-product zone—there is no zone for (non-existent) non-dairy cheese. This means that as the number of contours increases, Euler diagrams are typically less visually complex than the equivalent Venn diagram, particularly if the number of non-empty intersections is small.^{[19]}

The difference between Euler and Venn diagrams can be seen in the following example. Take the three sets:

The Euler and the Venn diagram of those sets are:

### Extensions to higher numbers of sets[edit]

Venn diagrams typically represent two or three sets, but there are forms that allow for higher numbers. Shown below, four intersecting spheres form the highest order Venn diagram that has the symmetry of a simplex and can be visually represented. The 16 intersections correspond to the vertices of a tesseract (or the cells of a 16-cell, respectively).

For higher numbers of sets, some loss of symmetry in the diagrams is unavoidable. Venn was keen to find "symmetrical figures...elegant in themselves,"^{[11]} that represented higher numbers of sets, and he devised an *elegant* four-set diagram using ellipses (see below). He also gave a construction for Venn diagrams for *any* number of sets, where each successive curve that delimits a set interleaves with previous curves, starting with the three-circle diagram.

Venn's construction for four sets

Venn's construction for five sets

Venn's construction for six sets

Venn's four-set diagram using ellipses

**Non-example:**This Euler diagram is*not*a Venn diagram for four sets as it has only 13 regions (excluding the outside); there is no region where only the yellow and blue, or only the red and green circles meet.Five-set Venn diagram using congruent ellipses in a five-fold rotationally symmetrical arrangement devised by Branko Grünbaum. Labels have been simplified for greater readability; for example,

**A**denotes**A**∩**B**^{c}∩**C**^{c}∩**D**^{c}∩**E**^{c}, while**BCE**denotes**A**^{c}∩**B**∩**C**∩**D**^{c}∩**E**.

### Edwards–Venn diagrams[edit]

Anthony William Fairbank Edwards constructed a series of Venn diagrams for higher numbers of sets by segmenting the surface of a sphere, which became known as Edwards–Venn diagrams.^{[20]} For example, three sets can be easily represented by taking three hemispheres of the sphere at right angles (*x* = 0, *y* = 0 and *z* = 0). A fourth set can be added to the representation, by taking a curve similar to the seam on a tennis ball, which winds up and down around the equator, and so on. The resulting sets can then be projected back to a plane, to give *cogwheel* diagrams with increasing numbers of teeth—as shown here. These diagrams were devised while designing a stained-glass window in memory of Venn.^{[20]}

### Other diagrams[edit]

Edwards–Venn diagrams are topologically equivalent to diagrams devised by Branko Grünbaum, which were based around intersecting polygons with increasing numbers of sides. They are also two-dimensional representations of hypercubes.

Henry John Stephen Smith devised similar *n*-set diagrams using sine curves^{[20]} with the series of equations

Charles Lutwidge Dodgson (also known as Lewis Carroll) devised a five-set diagram known as Carroll's square. Joaquin and Boyles, on the other hand, proposed supplemental rules for the standard Venn diagram, in order to account for certain problem cases. For instance, regarding the issue of representing singular statements, they suggest to consider the Venn diagram circle as a representation of a set of things, and use first-order logic and set theory to treat categorical statements as statements about sets. Additionally, they propose to treat singular statements as statements about set membership. So, for example, to represent the statement "a is F" in this retooled Venn diagram, a small letter "a" may be placed inside the circle that represents the set F.^{[21]}

### Related concepts[edit]

Venn diagrams correspond to truth tables for the propositions , , etc., in the sense that each region of Venn diagram corresponds to one row of the truth table.^{[22]}^{[23]} This type is also known as Johnston diagram. Another way of representing sets is with John F. Randolph's R-diagrams.

### See also[edit]

### Notes[edit]

**^**In Euler's*Lettres à une princesse d'Allemagne sur divers sujets de physique et de philosophie*[Letters to a German Princess on various physical and philosophical subjects] (Saint Petersburg, Russia: l'Academie Impériale des Sciences, 1768), volume 2, pages 95-126. In Venn's article, however, he suggests that the diagrammatic idea predates Euler, and is attributable to Christian Weise or Johann Christian Lange (in Lange's book*Nucleus Logicae Weisianae*(1712)).

### References[edit]

- ^
^{a}^{b}^{c}"Comprehensive List of Set Theory Symbols".*Math Vault*. 2020-04-11. Retrieved 2020-09-05. **^**"Intersection of Sets".*web.mnstate.edu*. Retrieved 2020-09-05.- ^
^{a}^{b}"Sets and Venn Diagrams".*www.mathsisfun.com*. Retrieved 2020-09-05. **^**Venn, John. "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings"(PDF).*Penn Engineering*.**^**"The Philosophical Magazine: A Journal of Theoretical Experimental and Applied Physics".*Taylor & Francis*. Retrieved 2021-08-06.- ^
^{a}^{b}Venn, John (July 1880). "I. On the Diagrammatic and Mechanical Representation of Propositions and Reasonings"(PDF).*The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science*. 5.**10**(59): 1–18. doi:10.1080/14786448008626877. Archived(PDF) from the original on 2017-05-16.[1][2] - ^
^{a}^{b}Venn, John (1880). "On the employment of geometrical diagrams for the sensible representations of logical propositions".*Proceedings of the Cambridge Philosophical Society*.**4**: 47–59. - ^
^{a}^{b}Sandifer, Ed (2003). "How Euler Did It"(PDF).*MAA Online*. The Mathematical Association of America (MAA). Retrieved 2009-10-26. - ^
^{a}^{b}Ruskey, Frank; Weston, Mark (2005-06-18). "A Survey of Venn Diagrams".*The Electronic Journal of Combinatorics*. - ^
^{a}^{b}^{c}Lewis, Clarence Irving (1918).*A Survey of Symbolic Logic*. Berkeley: University of California Press. - ^
^{a}^{b}Venn, John (1881).*Symbolic logic*. Macmillan. p. 108. Retrieved 2013-04-09. **^**Mac Queen, Gailand (October 1967).*The Logic Diagram*(PDF) (Thesis). McMaster University. Archived from the original(PDF) on 2017-04-14. Retrieved 2017-04-14. (NB. Has a detailed history of the evolution of logic diagrams including but not limited to the Venn diagram.)**^**Leibniz, Gottfried Wilhelm (1903) [ca. 1690]. "De Formae Logicae per linearum ductus". In Couturat, Louis (ed.).*Opuscules et fragmentes inedits de Leibniz*(in Latin). pp. 292–321.**^**Baron, Margaret E. (May 1969). "A Note on The Historical Development of Logic Diagrams".*The Mathematical Gazette*.**53**(384): 113–125. doi:10.2307/3614533. JSTOR 3614533.**^**Henderson, David Wilson (April 1963). "Venn diagrams for more than four classes".*American Mathematical Monthly*.**70**(4): 424–426. doi:10.2307/2311865. JSTOR 2311865.**^**Ruskey, Frank; Savage, Carla D.; Wagon, Stan (December 2006). "The Search for Simple Symmetric Venn Diagrams"(PDF).*Notices of the AMS*.**53**(11): 1304–1311.**^**"Strategies for Reading Comprehension Venn Diagrams". Archived from the original on 2009-04-29. Retrieved 2009-06-20.**^**Weisstein, Eric W. "Venn Diagram".*mathworld.wolfram.com*. Retrieved 2020-09-05.**^**"Euler Diagrams 2004: Brighton, UK: September 22–23". Reasoning with Diagrams project, University of Kent. 2004. Retrieved 2008-08-13.- ^
^{a}^{b}^{c}Edwards, Anthony William Fairbank (2004).*Cogwheels of the Mind: The Story of Venn Diagrams*. Baltimore, Maryland, USA: Johns Hopkins University Press. p. 65. ISBN .. **^**Joaquin, Jeremiah Joven; Boyles, Robert James M. (June 2017). "Teaching Syllogistic Logic via a Retooled Venn Diagrammatical Technique".*Teaching Philosophy*.**40**(2): 161–180. doi:10.5840/teachphil201771767. Archived from the original on 2018-11-21. Retrieved 2020-05-12.**^**Grimaldi, Ralph P. (2004).*Discrete and combinatorial mathematics*. Boston: Addison-Wesley. p. 143. ISBN .**^**Johnson, David L. (2001). "3.3 Laws".*Elements of logic via numbers and sets*. Springer Undergraduate Mathematics Series. Berlin, Germany: Springer-Verlag. p. 62. ISBN .

### Further reading[edit]

- Mahmoodian, Ebadollah S.; Rezaie, M.; Vatan, F. (March 1987). "Generalization of Venn Diagram"(PDF).
*Eighteenth Annual Iranian Mathematics Conference*. Tehran and Isfahan, Iran. Archived from the original(PDF) on 2017-05-01. Retrieved 2017-05-01. - Edwards, Anthony William Fairbank (1989-01-07). "Venn diagrams for many sets".
*New Scientist*.**121**(1646): 51–56. - Watkinson, John (1990). "4.10. Hamming distance".
*Coding for Digital Recording*. Stoneham, MA, USA: Focal Press. pp. 94–99, foldout in backsleeve. ISBN . (NB. The book comes with a 3-page foldout of a seven-bit cylindrical Venn diagram.) - Stewart, Ian (June 2003) [1992]. "Chapter 4. Cogwheels of the Mind".
*Another Fine Math You've Got Me Into*(reprint of 1st ed.). Mineola, New York, USA: Dover Publications, Inc. (W. H. Freeman). pp. 51–64. ISBN . - Glassner, Andrew (2004). "Venn and Now".
*Morphs, Mallards, and Montages: Computer-Aided Imagination*. Wellesley, MA, USA: A. K. Peters. pp. 161–184. ISBN . - Mamakani, Khalegh; Ruskey, Frank (2012-07-27). "A New Rose: The First Simple Symmetric 11-Venn Diagram". p. 6452. arXiv:1207.6452. Bibcode:2012arXiv1207.6452M. Archived from the original on 2017-05-01. Retrieved 2017-05-01.

### External links[edit]

Venn diagrams are a very commonly used graphing technique that illustrates levels of overlap between groups in data. They can be created in R using code written as part of the Bioconductor Project. We are following the directions supplied here for installing a package for linear models for microarray data ().

For more information on how to install packages from Bioconductor you can look at this page: https://bioconductor.org/install

Note that you need to install the latest version of R for this package to work properly!

To install the package we need to install package

Finally, we need to load this package.

We can now use the commands in this package for generating Venn diagrams. The data needed for a Venn diagram consists of a set of binary variables indicating membership. We will be using the hsb2 dataset consisting of data from 200 students including scores from writing, reading, and math tests. We will create indicators for “high” values in each of these variables and generate Venn diagrams that tell us about the degree of overlap in high math, writing, and reading scores.

Next, we can use the command to impose the structure needed to generate the Venn diagram.

a hw hm hr Counts [1,] 0 0 0 113 [2,] 0 0 1 18 [3,] 0 1 0 8 [4,] 0 1 1 8 [5,] 1 0 0 12 [6,] 1 0 1 8 [7,] 1 1 0 11 [8,] 1 1 1 22 attr(,"class") [1] "VennCounts"We can now generate our Venn diagram with the command:

While some of the options for the command are specific to tests run on microarray data, we can change some of the formatting. Below, we add names to the groups, we change the relative size of the labels and counts, and we opt for the counts to appear in red.

We could opt to present just two groups in this way, but it is not possible to add a fourth. Note that the size of the areas of overlap do not coincide with the relative counts. It is also worth noting that the areas in these Venn diagrams may suggest overlap where there, in fact, is none. The example below illustrates this.

- Nursing admission data form
- Spellbook ideas
- Worship tutorials
- Read chainsaw man
- Iphone xr setup problems

## Venn Diagram

The package allows to build Venn Diagrams thanks to its function. It takes as input a list of vector. Each vector providing words.

The function starts bycounting how many words are common between each pair of list. It then draws the result, showing each set as a circle.

The output is available as a file in your current working directory.

The function offers several option to customize the output. Those options allow to customize the circles, the set names, and the intersection values.

Here is an example showing the number of shared words in the lyrics of 3 famous french singers: (Nekfeu, Booba) and Georges Brassens.

This example is extensively described in this dedicated post.

Note the use of both the and options to make the circle border color different darker.

Related chart types

Grouped and Stacked barplot

Treemap

Doughnut

Pie chart

Dendrogram

Circular packing

## venn: Draw and display a Venn diagram

### Description

This function uses a variety of input data to draw and display a Venn diagram with up to 7 sets.### Usage

venn(x = "", size = 15, ilabels = FALSE, counts = FALSE, snames = c(""), zcolor = c(""), transparency = 1, ...)### Arguments

x

A single number (of sets), or a metacommand (see details), or a list of set values, or dataset containing boolean values.

size

Plot size, in centimeters.

ilabels

Logical, print the labels for each intersection.

counts

Logical, print the counts for each intersection.

snames

An optional parameter containing the names for each set.

zcolor

A vector of colors for the custom zones.

transparency

Transparency for the color specified with

...

Additional parameters, mainly for the outer borders of the sets

### Details

The argument can be either:- a single number of sets, between 1 and 7- a metacommand to draw custom intersection zones- a list of values for the different sets: each component is a set, and only the first 7 components are processed.- a dataset of boolean values.A "zone" is a union of set intersections. There are exactly intersections in a Venn diagram, where is the number of sets. To highlight an entire set, we need a union of all possible intersections which form that set. For example, in a 3 sets diagram, the (overall) first set is composed by four intersections: for what is in the first set but outside sets 2 and outside set 3 for the intersection between sets 1 and 3, outside set 2 for the intersection between sets 1 and 2, outside set 3 for the intersection between all three sets. A meta-language can be used to define these intersections, using the values of for what is inside the set, for what is outside the set, and when its either inside or outside of the set. The command is translated as "display only the first, entire set" is equivalent with the union of the four intersections . The parameter should have the same length as the number of sets specified by the parameter . When the parameter is used as a metacommand, the number of sets is calculated as the number of characters in each intersection of the metacommand. One such character command is or , and all intersections have exactly three characters. Argument is used when is a list or a dataframe. When a list, it counts the number of common values for each intersection. When is a dataframe (comprised of exclusively boolean values 0 and 1), it counts the number of similar rows, for each intersection. is activated by default when is a list or a dataframe. If is a number or a command, argument establishes the names for the different sets, or in its absence it is taken from . When is a list or a dataframe, is taken from their names. Via , users can specify additional parameters, mainly for the outer borders of the sets, as specified by . All of them are feeded to the function which is responsible with the borders. For up to 3 sets, the shapes can be circular. For more than 3 sets, the shape cannot be circular, in the case of 4 and 5 sets they are ellipses. For more than 5 sets the shapes cannot be continous (they might be monotone, but not continous). The 7 sets diagram is called "Adelaide" (Ruskey, 2005). The most challenging diagram is the one with 6 sets, where for many years it was thought a Venn diagram didn't even exist. All diagrams are symetric, except for the one with 6 sets, where some of the sets have different shapes. The diagram in this package is an adaptation from Mamakani, K., Myrvold W. and F. Ruskey (2011),### References

Ruskey, F. and M. Weston. 2005.*Venn diagrams*. Electronic Journal of Combinatorics, Dynamic Survey DS vol.5. Mamakani, K., Myrvold W. and F. Ruskey. 2011.

*Generating all Simple Convexly-drawable Polar Symmetric 6-Venn Diagrams*. International Workshop on Combinatorial Algorithms, Victoria. LNCS, 7056, 275-286.

### Examples

# A simple Venn diagram with 3 sets venn(3) # display the first whole set venn("1--") # an equivalent command venn("100 + 110 + 101 + 111") # another equivalent command venn(c("100","110","101","111")) # adding the labels for the intersections venn("1--", ilab = TRUE) # using different parameters for the borders venn(4, lty = 5, col = "navyblue") # a 5 sets Venn diagram venn(5) # a union of two sets venn("1---- + ----1") # with different colors venn("1---- + ----1", zcol = c("red", "blue"), transp = 0.3) # 6 sets diagram venn(6) # 7 sets "Adelaide" venn(7) # first set colored venn("1------", ilab = TRUE) # artistic version venn(c("1000000", "0100000", "0010000", "0001000", "0000100", "0000010", "0000001", "1111111")) # when x is a list set.seed(12345) x <- list(First = 1:20, Second = 10:30, Third = sample(25:50, 15)) venn(x) # when x is a dataframe set.seed(12345) x <- as.data.frame(matrix(sample(0:1, 150, replace=TRUE), ncol=5)) venn(x)## Diagram sets 6 venn r

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Plotting in R for Biologists -- Lesson 6: Plot anything!.

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