Set theory has applications in the real world, from bars to train schedules
ILLUSTRATION: TOMASZ WALENTZ
10 COMMENTS
Mathematics often helps us to think about issues that don’t seem mathematical. One area that has surprisingly far-reaching applications is the theory of sets. Sets are one of the most basic objects in mathematics, since we almost always have a collection of things we are trying to study.
But interesting sets are more than a random bunch of objects, just as a house is more than a pile of bricks. Mathematical structure arises from relationships, such as addition and multiplication, distance and closeness, or—in the case of sets—ordering.
An ordered set is based on a defined relationship among its objects. For example, when people line up at a store’s checkout counter, they are usually arrayed in an unambiguous order, based on who arrived first, second and so on. The people form an ordered set. By contrast, at many bars, customers have to crowd around until a bartender notices them. This is not an ordered set.
In mathematics, we often think about comparing sets using functions, which pair up the objects in one set with the objects in another. In a train timetable, for example, departure times are paired with the corresponding arrival times. When applying functions to ordered sets, it is revealing to ask whether or not the order is “preserved.” If a local train leaves earlier than an express train for the same destination but arrives later, the function does not preserve the order of the set.
When functions are not order-preserving in mathematics, it causes structural breakdown. In life, it can cause antagonism: At a bar, if people are not served in the order they arrived, they feel miffed. A more serious example is the late Sen. John McCain, who declined to accept early release as a prisoner of war unless those captured before him were released.
Trouble also can arise when one set of people is ordered according to two different rules. At a workplace, for instance, the employees can be arranged by age and also by the seniority of their positions. When the order is not preserved in moving from one set to the other—when, say, a young person is the superior of an older person—tensions may ensue.
Or consider a single set of people arranged by both wealth and some ostensible marker of privilege. Arguments can arise when the function comparing them does not preserve order—for example, if a white person is alleged to be more privileged than a nonwhite person, even though the nonwhite person is richer.
Orderings of sets are seldom so straightforward, of course. We can stack our Tupperwaredishes neatly if they fit inside each other, but some won’t fit if they are different shapes. In mathematics this is called a partially ordered set, in which some things are allowed to be “incomparable.” If we have multiple criteria for ordering a set, we might find one object higher up according to one rule, and another object higher up according to the other.
Thus, if you are looking to buy an apartment, you might find one that is in better condition on the inside although another has a much better location. This can cause indecision if you don’t know how to rank the criteria themselves in order of importance. You could make a strict hierarchy if you disregard all but one of the criteria, but it would be rather short-sighted to buy an apartment based on one criterion alone.
Partially ordered sets are more difficult to deal with but are also more common. They are just one of the many types of subtle mathematical relationships that have much to teach us about navigating the complexities of our own lives.
—Dr. Cheng’s new book, “The Art of Logic in an Illogical World,” will be published by Basic Books on Sept. 11.
