An Euler diagram is a diagrammatic means of representing sets and their relationships. The first use of "Eulerian circles" is commonly attributed to Swiss mathematician Leonhard Euler (1707–1783). They are closely related to Venn diagrams.
Venn 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 by other curriculum fields such as reading.[1]
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Euler diagrams consist of simple closed curves (usually circles) in the plane that depict sets. The sizes or shapes of the curves are not important, the significance of the diagram is in how they overlap. The spatial relationships between the regions bounded by each curve (overlap, containment or neither) corresponds to set-theoretic relationships (intersection, subset and disjointness).
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 the possible zones of overlap between its curves, representing all combinations of inclusion/exclusion of its constituent sets, but in an Euler diagram some zones might be missing. When the number of sets grows beyond 3, or even with three sets, but under the allowance of more than two curves passing at the same point, we start seeing the appearance of multiple mathematically unique Venn diagrams. Venn diagrams represent the relationships between n sets, with 2n zones, Euler diagrams may not have all zones. (An example is given below in the History section; in the top-right illustration the O and I diagrams are merely rotated; Venn stated that this difficulty in part led him to develop his diagrams).
In a logical setting, one can use model theoretic semantics to interpret Euler diagrams, within a universe of discourse. In the examples above, the Euler diagram depicts that the sets Animal and Mineral are disjoint since the corresponding curves are disjoint, and also that the set Four Legs is a subset of the set of Animals. The Venn diagram, which uses the same categories of Animal, Mineral, and Four Legs, does not encapsulate these relationships. Traditionally the emptiness of a set in Venn diagrams is depicted by shading in the region. Euler diagrams represent emptiness either by shading or by the use of a missing 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 diagram to the right, 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.
As shown in the illustration to the right, William Rowan Hamilton (1805–1865) in his 1860 asserts that the original use of circles to "sensualize ... the abstractions of Logic" (p. 180) was not Leonhard Paul Euler (1707–1783) but rather Christian Weise (?–1708) in his Nucleus Logicoe Weisianoe that appeared in 1712 posthumously. He references Euler's Letters to a German Princess on different Matters of Physics and Philosophy1" [1Partie ii., Lettre XXXV., ed. Cournot. – ED.][2]
In Hamilton's illustration the four forms of the syllogism as symbolized by the drawings A, E, I and O are:[3]
In his 1881 Symbolic Logic Chapter V "Diagrammatic Representation", John Venn (1834–1923) comments on the remarkable prevalence of the Euler diagram:
But nevertheless, he contended "the inapplicability of this scheme for the purposes of a really general Logic" (page 100) and in a footnote observed that "it fits in but badly even with the four propositions of the common Logic [the four forms of the syllogism] to which it is normally applied" (page 101). Venn ends his chapter with the observation that will be made in the examples below – that their use is based on practice and intuition, not on a strict algorithmic practice:
Finally, in his Chapter XX HISTORIC NOTES Venn gets to a crucial criticism (italicized in the quote below); observe in Hamilton's illustration that the O (Particular Negative) and I (Particular Affirmative) are simply rotated:
(Sandifer 2003 reports that Euler makes such observations too; Euler reports that his figure 45 (a simple intersection of two circles) has 4 different interpretations). Whatever the case, armed with these observations and criticisms, Venn then demonstrates (pp. 100–125) how he derived what has become known as his Venn diagrams from the "old-fashioned Euler diagrams". In particular he gives an example, shown on the left.
By 1914 Louis Couturat (1868–1914) had labeled the terms as shown on the drawing on the right. Moreover, he had labeled the exterior region (shown as a'b'c') as well. He succinctly explains how to use the diagram – one must strike out the regions that are to vanish:
Given the Venn's assignments, then, the unshaded areas inside the circles can be summed to yield the following equation for Venn's example:
In Venn the 0th term, x'y'z', i.e. the background surrounding the circles, does not appear. Nowhere is it discussed or labeled, but Couturat corrects this in his drawing. The correct equation must include this unshaded area shown in boldface:
In modern usage the Venn diagram includes a "box" that surrounds all the circles; this is called the universe of discourse or the domain of discourse.
Couturat now observes that, in a direct algorithmic (formal, systematic) manner, one cannot derive reduced Boolean equations, nor does it show how to arrive at the conclusion "No X is Z". Couturat concluded that the process "has ... serious inconveniences as a method for solving logical problems":
Thus the matter would rest until 1952 when Maurice Karnaugh (1924– ) would adapt and expand a method proposed by Edward W. Veitch; this work would rely on the truth table method precisely defined in Emil Post's 1921 PhD thesis "Introduction to a general theory of elementary propositions" and the application of propositional logic to switching logic by (among others) Claude Shannon, George Stibitz, and Alan Turing.[4] For example, in chapter "Boolean Algebra" Hill and Peterson (1968, 1964) present sections 4.5ff "Set Theory as an Example of Boolean Algebra" and in it they 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:
In Chapter 6, section 6.4 "Karnaugh Map Representation of Boolean Functions" they begin with:
The history of Karnaugh's development of his "chart" or "map" method is obscure. Karnaugh in his 1953 referenced Veitch 1951, Veitch referenced Claude E. Shannon 1938 (essentially Shannon's Master's thesis at M.I.T.), and Shannon 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 in Karnaugh map, Karnaugh in his method changed the order of the variables to correspond to what has become known as (the vertices of) a hypercube.
This example shows the Euler and Venn diagrams and Karnaugh map deriving and verifying the deduction "No X's are Z's". In the illustration and table the following logical symbols are used:
Given a proposed conclusion such as "No X is a Z", one can test whether or not it is a correct deduction by use of a truth table. The easiest method is put the starting formula on the left (abbreviate it as "P") and put the (possible) deduction on the right (abbreviate it as "Q") and connect the two with logical implication i.e. P → Q, read as IF P THEN Q. If the evaluation of the truth table produces all 1's under the implication-sign (→, the so-called major connective) then P → Q is a tautology. Given this fact, one can "detach" the formula on the right (abbreviated as "Q") in the manner described below the truth table.
Given the example above, the formula for the Euler and Venn diagrams is:
And the proposed deduction is:
So now the formula to be evaluated can be abbreviated to:
| Square # | 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 implication P → Q (i.e. ~(y & z) & (x → y) ) → ~(x & z) ) is still a formula, and the deduction – the "detachment" of Q out of P → Q – has not occurred. But given the demonstration that P → Q is tautology, the stage is now set for the use of the procedure of modus ponens to "detach" Q: "No X's are Z's" and dispense with the terms on the left.[5]
Modus ponens (or "the fundamental rule of inference"[6]) is often written as follows: The two terms on the left, "P → Q" and "P", are called premises (by convention linked by a comma), the symbol ⊢ means "yields" (in the sense of logical deduction), and the term on the right is called the conclusion:
For the modus ponens to succeed, both premises P → Q and P must be true. Because, as demonstrated above the premise P → Q is a tautology, "truth" is always the case no matter how x, y and z are valued, but "truth" will only be the case for P in those circumstances when P evaluates as "true" (e.g. rows 0 OR 1 OR 2 OR 6: x'y'z' + x'y'z + x'yz' + xyz' = x'y' + yz').[7]
One is now free to "detach" the conclusion "No X's are Z's", 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 "No X's are Z's"; the criterion for a successful deduction is that the 1's under the sub-major connective on the right include all the 1's under the sub-major connective on the left (the major 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 the all the same 1s that appear in the bold-faced column under the left-side sub-major connective & (rows 0, 1, 2 and 6), plus two more (rows 3 and 4).
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