December 11th, 2007

One of my favourite areas of mathematical study at University is known as Set Theory. Described at its simplest Set Theory allows one to create mathematical or logical relationships between group or sets of things.

Say you're helping to organize a party for your department and you've been delegated the task of making food arrangements and your task is finding a caterer that will provide something that everyone will enjoy.

While most would simply put two and two together like me you're fascinated by mathematics so you decide to solve your problem by creating a number of Venn Diagrams like the ones shown at right. For each Diagram A, B, and C, you've drawn a circle to represent the caterer and what they provide (Entity 1 which we'll represent as x) and another circle representing your associates and their eating preferences (Entity 2 which we'll call y). You then shade in various areas to help you determine what you're looking for.

Diagram A, you see, represents the intersection of x and y. This can be represented as a mathematical relationship (but unfortunately I can't make my web page editor represent it! Wah!). In plain English these are the meal options provided by the caterer that will satisfy some of your colleges.

Diagram B represents the union of x and y minus their intersection. This can be represented as a simple mathematical formula (xUy). In plain English these are the meals offered that no one is interested in eating.

Diagram B represents y and anywhere it overlaps x. Mathematically this can simply be represented as y. In plain English these are all the things your colleges would be interested in eating and which of those are offered by a given caterer.

What do you do? Oh, whatever do you do?!

Simplest solution?

Draw y inside x. In other words, find a caterer that offers something everyone wants to eat!

But what if you can't find a caterer that can provide that? Then what you need to do is find another caterer that can offer y - xUy or basically someone that can feed those that aren't fed by the first caterer.

The thing that attracted me to Set Theory is that it can be represented visually, mathematically, or, as above, in good old English. The useful aspect of Set Theory is that once you understand how to represent the problem visually the various pieces can be converted into straight forward mathematical representations which can be easily manipulated to quickly solve more complex problems (like arranging a banquet for 500 people from different countries all with extremely unique dietary needs).

As both someone who was trained in psychology and interested in interpersonal relationships I've often wondered if there are ways to represent human emotions and relationships in a manner that can be quantified then manipulated mathematically in such a way that would accurately represent the real world. Could one use Set Theory, for instance, to quantify, predict, and change emotional states? Or could it be used to identify or predict successful teams within an organization?

Another area of interest, as you probably already know, is spirituality, in particular Buddhism. Could Set Theory be used to predict or help bring about harmony? Could these mathematical representations demonstrate where there will be potential discord in our lives and how we can alleviate that?