We have two German mathematicians to thank for the development of modern set theory. Believe it, or not, this discipline is a fairly recent one, dating from the work of Georg Cantor (1845-1918) and Richard Dedekind (1831-1916) in the 1870s. Basically, sets are collections of objects and set theory is a form of mathematical logic that studies those sets and objects, their memberships in various sets, and the relationships between sets.
Almost immediately, a number of clever paradoxes were uncovered, one particularly famous one by Bertrand Russell (1872-1970). These paradoxes showed that the simple definitions of sets and membership of sets was too naïve and needed to be revised. To see why, consider Russell’s paradox.
Imagine that you sort through all possible sets and separate them into two categories … ones that are members of themselves and ones that are not members of themselves. For example, the set of all possible triangles is not a triangle itself. This means that this particular set does not contain itself. However, consider the set of all things that are not triangles. Obviously, that set is itself not a triangle, so it must be a member of itself. So far, so good. Now you have two very large sets. Consider the set that contains all the sets that are not members of themselves (like the set of triangles) and call it R. We ask a simple question, “Is R a member of itself?” If R is not a member of itself, then it must, by definition belong to the collection/set of sets that are not members of themselves … which is R itself! In other words, if it is not a member of itself, then it must be! Early set theory had to deal with a number of these surprisingly elementary but challenging issues before it developed as a powerful tool. Set theory is currently concerned with a very wide range of topics of great mathematical interest, including questions about the structure of the real number line and the consistency of sets/categories of numbers.
In the latter half of the 1800s, the English mathematician and author Charles Lutwidge Dodgson (1832-1898) had invented and used two-way tables to separate sets of items in a yes-no fashion. You would better know him as Lewis Carroll, the author of Alice in Wonderland and Through the Looking Glass. His tables were called Carroll Diagrams. In order to show more complex relationships, however, the English logician and philosopher, John Venn (1834-1923), invented Venn diagrams around the year 1880. They are capable of showing all possible logical relationships between a finite collection of different sets. They proved to be a very useful tool for illustrating elementary set theory as well as relationships in probability, logic, statistics, linguistics and computer science. As you can also see from the diagram above, they can be used in a creative way for fun! Viewing funny Venn Diagrams is a good way to learn about their structure and how to read them. Let me recommend that you Google ‘Funny Venn Diagrams’ and select [Images].
As I add material here, I will be explaining sets and their relationships to you (unions and intersections, for example) and introducing you to the symbolic ways in which these are expressed. We shall also solve some interesting problems using Venn Diagrams.
By far the best educator I have been subject to in my schooling life. Apart from showing me with the utmost clarity the many concepts of mathematics, Graeme has inspired me to better myself in everything that I do in school and in my life. His dedication to his craft is truly admirable, and this level of dedication to helping the student achieve is something that is extremely hard to find. If you’d want anyone on your side during HSC mathematics, it’d be Graeme.
Harry B (student, 2012)
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