Euler diagram


An Euler diagram is a diagrammatic means of representing sets 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 mathematician Leonhard Euler. In the United States, both Venn and Euler diagrams were incorporated as part of instruction in set theory as part of the new math movement of the 1960s. Since then, they have also been adopted by other curriculum fields such as reading as well as organizations and businesses.
Euler diagrams consist of simple closed shapes in a two dimensional plane that each depict a set or category. How or if these shapes overlap demonstrates the relationships between the sets. 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 that do not overlap represent disjoint sets, which have no elements in common. Two curves that overlap represent sets that intersect, that have common elements; the zone inside both curves represents the set of elements common to both sets. A curve completely within the interior of another is a subset of it.
Venn diagrams are a more restrictive form of Euler diagrams. A Venn diagram must contain all 2n logically possible zones of overlap between its n curves, representing all combinations of inclusion/exclusion of its constituent sets. Regions not part of the set are indicated by coloring them black, in contrast to Euler diagrams, where membership in the set is indicated by overlap as well as color.

History

As shown in the illustration to the right, Sir William Hamilton in his posthumously published Lectures on Metaphysics and Logic erroneously asserts that the original use of circles to "sensualize... the abstractions of Logic" was not Leonhard Paul Euler but rather Christian Weise in his Nucleus Logicae Weisianae that appeared in 1712 posthumously, however, the latter book was actually written by Johann Christian Lange rather than Weise. He references Euler's Letters to a German Princess
In Hamilton's illustration the four categorical propositions that can occur in a syllogism as symbolized by the drawings A, E, I and O are:
In his 1881 Symbolic Logic Chapter V "Diagrammatic Representation", John Venn 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" and on page 101 observed that, "It fits in but badly even with the four propositions of the common Logic to which it is normally applied." Venn ends his chapter with the observation illustrated 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 ; observe in Hamilton's illustration that the O and I are simply rotated:
. Whatever the case, armed with these observations and criticisms, Venn then demonstrates 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 had labeled the terms as shown on the drawing on the right. Moreover, he had labeled the exterior region 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 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 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 Claude Shannon, George Stibitz, and Alan Turing. For example, in chapter "Boolean Algebra", Hill and Peterson 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, 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 a hypercube.

Relation between Euler and Venn diagrams

s are a more restrictive form of Euler diagrams. A Venn diagram must contain all 2n logically possible zones of overlap between its n curves, representing all combinations of inclusion/exclusion of its constituent sets. Regions not part of the set are indicated by coloring them black, in contrast to Euler diagrams, where membership in the set is indicated by overlap as well as color. When the number of sets grows beyond 3 a Venn diagram becomes visually complex, especially compared to the corresponding Euler diagram. The difference between Euler and Venn diagrams can be seen in the following example. Take the three sets:
The Euler and the Venn diagrams of those sets are:
In a logical setting, one can use model theoretic semantics to interpret Euler diagrams, within a universe of discourse. In the examples below, 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 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.

Example: Euler- to Venn-diagram and Karnaugh map

This example shows the Euler and Venn diagrams and Karnaugh map deriving and verifying the deduction "No Xs are Zs".
In the illustration and table the following logical symbols are used:
: " 'It is not the case that: AND AND 'If an then a ". Once the propositions are reduced to symbols and a propositional formula &, one can construct the formula's truth 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 one can "reduce" the example's Boolean equation i.e. + 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 "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 and put the deduction on the right 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 1s under the implication-sign then P → Q is a tautology. Given this fact, one can "detach" the formula on the right 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:
At this point the above implication P → Q & → ~ 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 Xs are Zs" and dispense with the terms on the left.
Modus ponens is often written as follows: The two terms on the left, P → Q and P, are called premises, the symbol ⊢ means "yields", 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" is only the case for P in those circumstances when P evaluates as "true".
One is now free to "detach" the conclusion "No Xs are Zs", perhaps to use it in a subsequent deduction.
The use of tautological implication means that other possible deductions exist besides "No Xs are Zs"; the criterion for a successful deduction is that the 1s under the sub-major connective on the right include all the 1s under the sub-major connective on the left. For example, in the truth table, on the right side of the implication 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 &, plus two more.

Gallery