Graphical set representation involving overlapping circles
This article is about Eulerian circles of set theory and logic. For the geometric Euler circle, seeNine-point circle.
Euler diagram illustrating that the set of "animals with four legs" is a subset of "animals", but the set of "minerals" is a disjoint set (it has no members in common) with "animals"Euler diagram showing the relationships between differentSolar System objects
AnEuler diagram (/ˈɔɪlər/,OY-lər) is adiagrammatic means of representingsets and their relationships. They are particularly useful for explaining complex hierarchies and overlapping definitions. They are similar to another set diagramming technique,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 mathematicianLeonhard Euler (1707–1783). In the United States, both Venn and Euler diagrams were incorporated as part of instruction inset theory as part of thenew math movement of the 1960s. Since then, they have also been adopted by other curriculum fields such as reading[1] 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 theelements of the set, and the exterior, which represents all elements that are not members of the set. Curves which do not overlap representdisjoint sets, which have no elements in common. Two curves that overlap represent sets thatintersect, that have common elements; the zone inside both curves represents the set of elements common to both sets (theintersection of the sets). A curve completely within the interior of another is asubset 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 itsn 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.
A page from Hamilton'sLectures on Logic; the symbolsA,E,I, andO refer to four types of categorical statement which can occur in asyllogism (seedescriptions, left) The small text to the left erroneously says: "The first employment of circular diagrams in logic improperly ascribed to Euler. To be found in Christian Weise", a book which was actually written by Johann Christian Lange.[2][3]The diagram to the right is from Couturat[4](p 74) in which he labels the 8 regions of the Venn diagram. The modern name for the "regions" isminterms. They are shown in the diagram with the variablesx,y, andz per Venn's drawing. The symbolism is as follows: logicalAND [& ] is represented by arithmetic multiplication, and the logicalNOT [¬ ] is represented by ⟨′⟩ after the variable, e.g. the regionx′y′z is read as "(NOTx)AND (NOTy)ANDz" i.e. (¬x) & (¬y) &z .Both the Veitch diagram and Karnaugh map show all theminterms, but the Veitch is not particularly useful for reduction of formulas. Observe the strong resemblance between the Venn and Karnaugh diagrams; the colors and the variablesx,y, andz are per Venn's example.
As shown in the illustration to the right,Sir William Hamilton erroneously asserted that the original use of the circles to "sensualize... the abstractions of logic"[5] was notEuler (1707–1783) but ratherWeise (1642–1708);[6] however the latter book was actually written by Johann Christian Lange, rather than Weise.[2][3] He references Euler'sLetters to a German Princess.[7][a]
Venn (1834–1923) comments on the remarkable prevalence of the Euler diagram:
"... of the first sixty logical treatises, published during the last century or so, which were consulted for this purpose–somewhat at random, as they happened to be most accessible–it appeared that thirty four appealed to the aid of diagrams, nearly all of these making use of theEulerian scheme."[9]
Composite of two pages fromVenn (1881a), pp. 115–116 showing his example of how to convert a syllogism of three parts into his type of diagram; Venn calls the circles "Eulerian circles"[10]
But nevertheless, he contended, "the inapplicability of this scheme for the purposes of a really general logic"[9](p 100) and then noted that,
“It fits in, but badly, even with the four propositions of the common logic to which it is normally applied.”[9](p 101)
Venn ends his chapter with the observation illustrated in the examples below—that their use is based on practice and intuition, not on a strictalgorithmic practice:
“In fact ... those diagrams not only do not fit in with the ordinary scheme of propositions which they are employed to illustrate, but do not seem to have any recognized scheme of propositions to which they could be consistently affiliated.”[9](pp 124–125)
Finally, in his Venn gets to a crucial criticism (italicized in the quote below); observe in Hamilton's illustration that theO (Particular Negative) andI (Particular Affirmative) are simply rotated:
“We now come to Euler's well-known circles which were first described in hisLettres a une Princesse d'Allemagne (Letters 102–105).[7](pp 102–105) The weak point about these consists in the fact that they only illustrate in strictness the actual relations of classes to one another, rather than the imperfect knowledge of these relations which we may possess, or wish to convey, by means of the proposition. Accordingly they will not fit in with the propositions of common logic, but demand the constitution of a new group of appropriate elementary propositions. ... This defect must have been noticed from the firstin the case of the particular affirmative and negative, for the same diagram is commonly employed to stand for them both, which it does indifferently well”.[italics added][11][9](p 100, Footnote 1)[b]
Whatever the case, armed with these observations and criticisms, Venn[9](pp 100–125) then demonstrates how he derived what has become known as hisVenn diagrams from the “... old-fashioned Euler diagrams.” In particular Venn gives an example, shown at the left.
By 1914,Couturat (1868–1914) had labeled the terms as shown on the drawing at the right.[4] Moreover, he had labeled theexterior region (shown asa′b′c′) as well. He succinctly explains how to use the diagram – one muststrike out the regions that are to vanish:
"Venn's method is translated in geometrical diagrams which represent all the constituents, so that, in order to obtain the result, we need onlystrike out (by shading) those which are made to vanish by the data of the problem."[italics added][4](p 73)
Given the Venn's assignments, then, the unshaded areasinside the circles can be summed to yield the following equation for Venn's example:
"NOy isz andALLx isy: thereforeNOx isz" has the equationx′yz′ +xyz′ +x′y′z for the unshaded areainside the circles (but this is not entirely correct; see the next paragraph).
In Venn the background surrounding the circles, does not appear: That is, the term marked "0",x′y′z′ . Nowhere is it discussed or labeled, but Couturat corrects this in his drawing.[4] The correct equation must include this unshaded area shown in boldface:
"NOy isz andALLx isy: thereforeNOx isz" has the equationx′yz′ +xyz′ +x′y′z +x′y′z′.
In modern use, the Venn diagram includes a "box" that surrounds all the circles; this is called the universe of discourse or thedomain of discourse.
Couturat[4] observed that, in a directalgorithmic (formal, systematic) manner, one cannot derive reduced Boolean equations, nor does it show how to arrive at the conclusion "NOx isz". Couturat concluded that the process "has ... serious inconveniences as a method for solving logical problems":
"It does not show how the data are exhibited by canceling certain constituents, nor does it show how to combine the remaining constituents so as to obtain the consequences sought. In short, it serves only to exhibit one single step in the argument, namely the equation of the problem; it dispenses neither with the previous steps, i. e., "throwing of the problem into an equation" and the transformation of the premises, nor with the subsequent steps, i. e., the combinations that lead to the various consequences. Hence it is of very little use, inasmuch as the constituents can be represented by algebraic symbols quite as well as by plane regions, and are much easier to deal with in this form."[4](p 75)
Thus the matter would rest until 1952 whenMaurice Karnaugh (1924–2022) would adapt and expand a method proposed byEdward W. Veitch; this work would rely on thetruth table method precisely defined byEmil Post[12] and the application of propositional logic to switching logic by (among others)Shannon,Stibitz, andTuring.[c]For example, Hill & Peterson (1968)[13] present the Venn diagram with shading and all. They give examples of Venn diagrams to solve example switching-circuit problems, but end up with this statement:
"For more than three variables, the basic illustrative form of the Venn diagram is inadequate. Extensions are possible, however, the most convenient of which is the Karnaugh map, to be discussed in Chapter 6."[13](p 64)
In Chapter 6, section 6.4 "Karnaugh map representation of Boolean functions" they begin with:
"The Karnaugh map1 [1Karnaugh 1953] is one of the most powerful tools in the repertory of the logic designer. ... A Karnaugh map may be regarded either as a pictorial form of a truth table or as an extension of the Venn diagram."[13](pp 103–104)
The history of Karnaugh's development of his "chart" or "map" method is obscure. The chain of citations becomes an academic game of "credit, credit; ¿who's got the credit?":Karnaugh (1953) referencedVeitch (1952), Veitch, referencedShannon (1938),[14] andShannon (1938), in turn referenced (among other authors of logic texts)Couturat (1914). In Veitch's method the variables are arranged in a rectangle or square; as described inKarnaugh map, Karnaugh in his method changed the order of the variables to correspond to what has become known as (the vertices of) ahypercube.
Examples of smallVenn diagrams(on left) with shaded regions representingempty sets, showing how they can be easily transformed into equivalent Euler diagrams(right)
Venn diagrams are a more restrictive form of Euler diagrams. A Venn diagram must contain all 2n logically possible zones of overlap between itsn 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:
The Euler and the Venn diagrams of those sets are:
Euler diagram
Venn diagram
In a logical setting, one can use model-theoretic semantics to interpret Euler diagrams, within auniverse of discourse. In the examples below, the Euler diagram depicts that the setsAnimal andMineral are disjoint since the corresponding curves are disjoint, and also that the setFour Legs is a subset of the set ofAnimals. The Venn diagram, which uses the same categories ofAnimal,Mineral, andFour Legs, does not encapsulate these relationships. Traditionally theemptiness of a set in Venn diagrams is depicted by shading in the region. Euler diagrams representemptiness either by shading or by the absence of a region.
Often a set of well-formedness conditions are imposed; these are topological or geometric constraints imposed on the structure of the diagram. 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.
This example shows the Euler and Venn diagrams and Karnaugh map deriving and verifying the deduction "NoXs areZs".In the illustration and table the following logical symbols are used:
1 can be read as "true", 0 as "false"
~ for NOT and abbreviated to ′ when illustrating the minterms e.g. x′ =defined NOT x,
& (logical AND) between propositions; in the minterms AND is omitted in a manner similar to arithmetic multiplication: e.g. x′y′z =defined ~x & ~y & z (From Boolean algebra: 0⋅0 = 0, 0⋅1 = 1⋅0 = 0, 1⋅1 = 1, where "⋅" is shown for clarity)
→ (logical IMPLICATION): read as IF ... THEN ..., or " IMPLIES ",P →Q = defined NOTP ORQ
Before it can be presented in a Venn diagram or Karnaugh Map, the Euler diagram's syllogism "NoY isZ, AllX isY" must first be reworded into the more formal language of thepropositional calculus: " 'It is not the case that:Y ANDZ′ AND 'If anX then aY′ ". Once the propositions are reduced to symbols and a propositional formula ( ~(y & z) & (x → y) ), one can construct the formula'struth table; from this table the Venn and/or the Karnaugh map are readily produced. By use of the adjacency of "1"s in the Karnaugh map (indicated by the grey ovals around terms 0 and 1 and around terms 2 and 6) one can "reduce" the example'sBoolean equation i.e. (x′y′z′ + x′y′z) + (x′yz′ + xyz′) to just two terms: x′y′ + yz′. But the means for deducing the notion that "No X is Z", and just how the reduction relates to this deduction, is not forthcoming from this example.
Given a proposed conclusion such as "NoX is aZ", one can test whether or not it is a correctdeduction by use of atruth table. The easiest method is put the starting formula on the left (abbreviate it asP) and put the (possible) deduction on the right (abbreviate it asQ) and connect the two withlogical implication i.e.P →Q, read as IFP THENQ. If the evaluation of the truth table produces all 1s under the implication-sign (→, the so-calledmajor connective) thenP →Q is atautology. Given this fact, one can "detach" the formula on the right (abbreviated asQ) in the manner described below the truth table.
Given the example above, the formula for the Euler and Venn diagrams is:
So now the formula to be evaluated can be abbreviated to:
( ~(y & z) & (x → y) ) → ( ~ (x & z) ):P →Q
IF ( "NoYs areZs" and "AllXs areYs" ) THEN ( "NoXs areZs" )
The Truth Table demonstrates that the formula ( ~(y & z) & (x → y) ) → ( ~ (x & z) ) is a tautology as shown by all 1s in yellow column.
Square no.
Venn, Karnaugh region
x
y
z
(~
(y
&
z)
&
(x
→
y))
→
(~
(x
&
z))
0
x′y′z′
0
0
0
1
0
0
0
1
0
1
0
1
1
0
0
0
1
x′y′z
0
0
1
1
0
0
1
1
0
1
0
1
1
0
0
1
2
x′yz′
0
1
0
1
1
0
0
1
0
1
1
1
1
0
0
0
3
x′yz
0
1
1
0
1
1
1
0
0
1
1
1
1
0
0
1
4
xy′z′
1
0
0
1
0
0
0
0
1
0
0
1
1
1
0
0
5
xy′z
1
0
1
1
0
0
1
0
1
0
0
1
0
1
1
1
6
xyz′
1
1
0
1
1
0
0
1
1
1
1
1
1
1
0
0
7
xyz
1
1
1
0
1
1
1
0
1
1
1
1
0
1
1
1
At this point the above implicationP →Q (i.e. ~(y & z) & (x → y) ) → ~(x & z) ) is still a formula, and the deduction – the "detachment" ofQ out ofP →Q – has not occurred. But given the demonstration thatP →Q is tautology, the stage is now set for the use of the procedure ofmodus ponens to "detach" Q: "NoXs areZs" and dispense with the terms on the left.[nb 1]
Modus ponens (or "the fundamental rule of inference"[15]) is often written as follows: The two terms on the left,P →Q andP, are calledpremises (by convention linked by a comma), the symbol ⊢ means "yields" (in the sense of logical deduction), and the term on the right is called theconclusion:
P →Q,P ⊢Q
For the modus ponens to succeed, both premisesP →Q andP must betrue. Because, as demonstrated above the premiseP →Q is a tautology, "truth" is always the case no matter how x, y and z are valued, but "truth" is only the case forP in those circumstances whenP evaluates as "true" (e.g. rows0 OR1 OR2 OR6: x′y′z′ + x′y′z + x′yz′ + xyz′ = x′y′ + yz′).[nb 2]
i.e.: IF "NoYs areZs" and "AllXs areYs"THEN "NoXs areZs", "NoYs areZs" and "AllXs areYs" ⊢ "NoXs areZs"
One is now free to "detach" the conclusion "NoXs areZs", perhaps to use it in a subsequent deduction (or as a topic of conversation).
The use of tautological implication means that other possible deductions exist besides "NoXs areZs"; the criterion for a successful deduction is that the 1s under the sub-major connective on the rightinclude all the 1s under the sub-major connective on the left (themajor connective being the implication that results in the tautology). For example, in the truth table, on the right side of the implication (→, the major connective symbol) the bold-face column under the sub-major connective symbol "~ " has all the same 1s that appear in the bold-faced column under the left-side sub-major connective& (rows0,1,2 and6), plus two more (rows3 and4).
Euler diagram categorizing different types ofmetaheuristics
Euler Diagram displaying the relationship between homographs, homophones, and synonyms
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; some are even equivalent to Venn diagrams. Areas are shaded to indicate that they contain no elements.
Henri Milne-Edwards's (1844) diagram of relationships of vertebrate animals, illustrated as a series of nested sets
^By the time these lectures of Hamilton were published, Hamilton had died. His editors (marked byED.), responsible for most of the footnote text, were the logiciansHenry Longueville Mansel andJohn Veitch.
^Sandifer (2004) points out thatEuler himself also makes such observations: Euler reports that his figure 45 (a simple intersection of two circles) has 4 different interpretations.
^This is a sophisticated concept. Russell and Whitehead (2nd edition 1927) in theirPrincipia Mathematica describe it this way: "The trust in inference is the belief that if the two former assertions [the premises P, P→Q ] are not in error, the final assertion is not in error . . . An inference is the dropping of a true premiss [sic]; it is the dissolution of an implication" (p. 9). Further discussion of this appears in "Primitive Ideas and Propositions" as the first of their "primitive propositions" (axioms): *1.1 Anything implied by a true elementary proposition is true" (p. 94). In a footnote the authors refer the reader back to Russell's 1903Principles of Mathematics §38.
^Reichenbach discusses the fact that the implicationP →Q need not be a tautology (a so-called "tautological implication"). Even "simple" implication (connective or adjunctive) work, but only for those rows of the truth table that evaluate as true, cf Reichenbach 1947:64–66.
^abMac Queen, Gailand (October 1967).The Logic Diagram(PDF) (Thesis).McMaster University. p. 5. Archived fromthe original(PDF) on 2017-04-14. Retrieved2017-04-14. (NB. Has a detailed history of the evolution of logic diagrams including but not limited to the Euler diagram.)
Couturat, Louis (1914).The Algebra of Logic: Authorized English Translation by Lydia Gillingham Robinson with a Preface by Philip E. B. Jourdain. Chicago and London:The Open Court Publishing Company.
Jevons, W. Stanley (1880).Elementary Lessons in Logic: Deductive and Inductive. With Copious Questions and Examples, and a Vocabulary of Logical Terms. London and New York:M. A. MacMillan and Co.
Emil Post 1921 "Introduction to a general theory of elementary propositions" reprinted with commentary byJean van Heijenoort in Jean van Heijenoort, editor 1967From Frege to Gödel: A Source Book of Mathematical Logic, 1879–1931,Harvard University Press, Cambridge, MA,ISBN0-674-32449-8 (pbk.)
Claude E. Shannon 1938 "A Symbolic Analysis of Relay and Switching Circuits",Transactions American Institute of Electrical Engineers vol 57, pp. 471–495. Derived fromClaude Elwood Shannon: Collected Papers edited by N.J.A. Solane and Aaron D. Wyner,IEEE Press, New York.
