Links to Related Articles: S. Arlinghaus, W. Drake, J. Nystuen, A. Laug, K. Oswalt, D. Sammataro, Animaps.

 The Neglected Relation Sandra L. Arlinghaus, The University of Michigan  William C. Arlinghaus, Lawrence Technological University      Mathematics, like musical composition and other fine arts, is purely a human creation.  Without us, does it exist?  This sort of "meta" question has long interested scholars with multidisciplinary interests (readers are referred to the references section at the end for a few of the numerous references to ideas of this sort that have appeared in the literature over centuries).  Indeed, does the societal culture in which the predominant mathematics is developed and embedded in a particular historical epoch influence the kind of mathematics that is developed?  Again, this question has been studied in many ways:  we consider one case here--that of the mathematical relation and selected real-world interpretations.  These are displayed in a number of visual formats not merely as curiosities but more significantly for the suggestion they might offer as to why or why not certain types of formal structures get created.  It is important to attempt to understand deeper processes such as these:  the mathematics we use in the real-world often influences the decisions we make.       Municipal authorities might use demographic forecasts based on curve fitting to guide the direction of urban land use planning.  A City Administrator might use a rank-ordered set of priorities to decide how valuable taxpayer funds will be allocated over a period of years to develop (or not develop) infrastructure.  The way in which the mathematics is used will influence the outcome of the analysis and, therefore quite likely, the policy that is set in place.  We also invite feedback, using the capability of the internet, so that readers might share other cases with each other (near the end of this document).

 One-to-many transformations between two mathematical sets are an often neglected class of relationships. Much of modern mathematics, for example, considers only functions.  A function, mapping a set X to a set Y permits an element x in X to be sent to an element y in Y, or it permits a number of elements, x1, x2, x3 in X to be sent to a single element y in Y (Figure 1, left side).  The former situation is one-to-one and the latter is many-to-one.  Functions may be one-to-one or they may be many-to-one; they may not be one-to-many.  They are "single-valued."  Graphically, the idea is represented as in Figure 1.  When one element of X is permitted to map to many elements of Y, as in x mapping to y1, y2, y3 (Figure 1, right half) the associated mathematical transformation is often referred to as a relation.

 Figure 1.  A function requires that each element of X be associated with only one element of Y (although many different elements of X may be associated with the same element of Y).  A relation removes this restriction, allowing one element of X to be associated with many elements of Y.  Thus, every function is relation, but not every relation is a function.

 In the Cartesian coordinate system the same idea may be visualized as in Figure 2.  In the case of a function, a vertical line cuts the graph of the function no more than once (Figure 2a shows a one-to-one function and Figure 2b shows a many-to-one function).  In a graph that is not a function (not "single-valued"), the vertical line may cut this curve (that is not the graph of a function) in more than one place (Figure 2c shows such a graph).

 Figure 2a.  One-to-one function. Figure 2b.  Many-to-one function.  Many x-values (green dots) correspond to one y-value (height of horizontal line). Figure 2c.  One-to-many relation. One x-value (green dot) corresponds to many y-values (red dots)

 The visual display of the difference between function and relation, the many-to-one and the one-to-many, is clear in the Cartesian coordinate system because the ordering of the function from X to Y is clear in our minds.  In a coordinate-free environment, such as the world of the Applet (TM, Sun Microsystems), all that is evident is the structural equivalence of many-to-one and one-to-many transformations (Figure 3).  In Figure 3, note the stability of the one-to-one transformation as the graphic moves; the many-to-one and the one-to-many never quite settle down to a totally stable configuration.  This lack is a function of pattern involving length of edges joining nodes and dimension of the square universe of discourse in which the Applets (TM Sun Microsystems) live.  In the case of Figure 3, it may simply be a function of a particular commensurability pattern of edges and underlying raster; nonetheless, the general consideration as to what sorts of configurations exhibit geometric stability is an important one, particularly as in regard for looking for points of intervention into process (see varroa mite mapplet).  K. Sims has noted the importance of such lack of stability in anthropological contexts building on island networks found in Hage and Harary.

 Figure 3.  Applets (TM, Sun Microsystems) show one-to-one, many-to-one, and one-to-many transformations.  Note the structural equivalence between the many-to-one and the one-to-many applets.

 The relation is often ignored in mathematical analyses of various sorts.  Perhaps that is because the definite nature of single-valued mappings is regarded as important.  Is the world, however, single-valued?  We consider a few real-world situations in which relations can be observed to be the underlying conceptual force. Postal Transformation      A simple, convenient example often given to students studying functions for the first time is the following postal example.   Given a set of hard-copy handwritten letters in envelopes that are to be sent through the conventional U.S. Postal Service network by regular first-class mail. My letter can be sent to a single address (one-to-one). My set of three different letters can be sent to a single address (many-to-one). My one letter cannot go to three different addresses (not one-to-many).      Some might argue that the invention of the printing press permitted one page to go to many.  Yet, there is variation from page to page--there are ink splatters, broken type, and so forth.      Still others might assert that photocopying of a page will enable one letter to go many different addresses, as long as the original as distinct from the rest is not included--hence the rise of junk mail.  Someone else might argue, however, that any two photocopies differ from each other on account of diminishing the amount of toner available for copies later in the process.       Further, if one considers virtual messages, rather than hard copy messages, then a single e-letter can be sent to a single address (one-to-one), a set of three different e-notes can be sent to a single address (many-to-one), and a single note can go simultaneously to three different addresses (one-to-many).  The electronic revolution of our "information age" offers a true postal transformation from the functional to the relational.       Perhaps a common theme in all these refinements of argument will be that to move from one style of mathematical transformation to another in the real-world requires some sort of underlying real-world transformation through invention, revolution, or other remarkable event.  Hence, the argument for the printing press, the photocopying machine, and the e-mail/computer all have merit.  Indeed, Solstice, itself, takes advantage of this one-to-many relational capability!

 Home Ownership      As we look around our environment today, of midwestern United States of America, we see a variety of dwelling types and of ownership of them. One-to-one ownership:  one family owns a single parcel of land, often in a suburban area and elsewhere when land is plentiful and land values are relatively low. Many-to-one ownership:  many families own a single parcel or building.  This style of ownership is often "condominium" or "cooperative" ownership.  In the landscape it is evident mostly in more densely populated areas or where land values are relatively high. One-to-many ownership:  one family owns many residences.  This particular situation, not represented as a mathematical function but only as a relation, is perhaps not as common as the two above.  Typically, one might expect families with excess wealth to own more than one residence.  Our colleague John Nystuen asked where such individuals cast votes.  We explore the dynamics of that situation below.

 Composition of Transformations      If one were to map the relations listed above for home ownership, a figure similar to Figure 3 would be the result.  When voting is added on, the situation becomes more complicated, given that voting is done and counted locally and not nationally. In the one-to-one situation, the homeowner registers to vote from his or her single address and there is no difficulty counting the vote. In the many-to-one situation, the homeowners register to vote from their single address and there is no difficulty counting the vote.  All go to the same polling place to vote. In the one-to-many situation, however, a person who owns property in Michigan and in Florida, for example, might attempt to vote in two places even though he/she is only entitled to one vote.  Figure 4 shows that when only a single vote is cast, as it should be, the system remains closed, bounded, and manageable (in some sense).  When more than one vote is cast, the system may rapidly fall out of order, especially when there are thousands or hundreds of thousands of people who own more than one residence from which they might vote.  Some sort of nationalized database on voter registration and residency might make the problem more tractable. What is important in this case is the composition of mappings:  one followed by another.  In this case, the two mappings are home ownership followed by voting.  When the first is a function, the composition works well in the real-world interpretation.  When it is only a relation, there is room for serious manipulation that was not present in the functional characterization.  Extra care is appropriate when composing mappings.

 Figure 4a.  Voter x owns three residences, y1, y2, and y3 and casts the one legal vote, z, to which he/she is entitled.  Voter a owns three residences, b1, b2, and b3 and casts one legal vote c1 (from residence b1) and two illegal votes (from residences b2 and b3), c2, and c3.  Note that the legal case is visually manageable in some sense while the illegal case sprawls across the map and is more difficult to track. Figure 4b.  When  homeownership and voting become more complicated, the closure and sprawl noted above (in the caption to Figure 4a) become more evident.

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 References Arlinghaus, Sandra L.; Arlinghaus, William C.; and, Harary, Frank (2001), Graph Theory and Geography:  An Interactive View.  New York:  John Wiley and Sons. Arlinghaus, Sandra L.; Drake, William D.; Nystuen, John D.; Laug, Audra; Oswalt, Kris; Sammataro, Diana (1998),  Animaps, Solstice:  An Electronic Journal of Geography and Mathematics, Volume IX, Number 1.  Ann Arbor:  Institute of Mathematical Geography. Gershenson, Daniel E. (1964), Anaxagoras and the Birth of Scientific Method.  New York:  Blaisdell Publishing Company. Hage, Per and Harary, Frank (1996), Island Networks:  Communication, Kinship, and Classification Structures in Oceania.  Cambridge University Press. Jeans, James H. (1929), Eos, or, The Wider Aspects of Cosmogony.  New York:  E. P. Dutton and Company. Kuhn, Thomas S. (1957), The Copernican Revolution:  Planetary Astronomy in the Development of Western Thought. Cambridge, MA:  Harvard University Press. Kuhn, Thomas S. (1962), The Structure of Scientific Revolutions. Chicago:  University of Chicago Press. Quine, Willard Van Orman (1969). Ontological Relativity, and Other Essays. New York: Columbia University Press.  Sims, K.  The Fiji Islands and the Concept of Spatial Hierarchy, unpublished, 2001.