Set Theory

Grouping objects, whether they are tangible objects such as cars, books or animals or intangible objects like colors or numbers is not hard to do. There are in fact many ways to do so, the most familiar being the way that most elementary students learn. Venn diagrams are usually one of the first things we learn about set theory. Basic Venn diagrams are normally drawn as circles that overlap. If the first circle (we’ll call it “A”) represents the group (set) of insects that sting and the second circle (circle “B”) represents the group (set) of insects that fly then all of the insects that both fly and sting would be represented by the overlapping part of the circles.

When done on math, grouping objects is known as Set Theory. Sets are represented in a different way but it is still the same concept. You can define a set to be a group of actual object or you can define a set to by a specific rule such as “set A contains all even numbers”. The objects in a set are called “elements” and the operations of sets are quite simple, the most common being the union, intersection and difference. The union is simply the set of elements that contain any elements of set A, B or both. The intersection is the set of elements that set A and B both have in common while the difference is the set of elements that are in A but not in B. Using Venn diagrams as an example, if we highlight the areas of a circle that is the union of A and B then both circles would be completely highlighted. For the intersection the area of the circles that overlap would be the area that is highlighted and for the difference the area of the circle that would be highlighted is the part of circle A that is not overlapping with B.

Venn diagram representation of a union.

Venn diagram representation of a union.

Venn diagram representation of an intersection.

Venn diagram representation of an intersection.

Venn diagram representation of a difference.

Venn diagram representation of a difference.

Although named for him, John Venn did not invent these diagrams; logicians have used them for centuries. It was common in the 19th century to use Euler diagrams (Eulerian circles). Euler diagrams consisted mainly of circles within circles and occasionally circles by themselves. As an example, if the outer circle represented insects that sting, then the circle inside of that would represent insects that both sting and fly. A completely separate circle would represent something that neither flew nor stung. John Venn felt that theses diagrams were inadequate and reverted back to a diagram that has been used throughout history. Since Venn formalized these diagrams and was the first to generalize them, they were later named after him.

It is interesting to note that the original purpose for Venn diagrams was not set theory but rather symbolic logic. Symbolic logic uses symbols rather than words in order to remove the ambiguity that some words tend to have. When using abstract symbols rather than familiar words, it is harder to see the truth of a statement. Venn diagrams helped greatly with this. In symbolic logic you have two premises and a conclusion.

Most mathematical topics normally develop through the collaboration of many mathematicians, but a single mathematician, Georg Cantor, founded set theory in the late nineteenth century. There are many different subfields of set theory including Combinatorial set theory, Descriptive set theory, Fuzzy set theory and Rough set theory, but the one that is most widely known among mathematicians is Zermelo-Fraenkel set theory (ZFC). ZFC was originally developed in an attempt to rid set theory of paradoxes such as Russell’s Paradox, discovered in 1901 by Bertrand Russell. Russell’s Paradox can be stated as such: Let set R be the set of all sets that are not members of themselves. If R is not a member of itself, then by definition it must contain itself. But this contradicts its own definition of being the set of all sets that are not members of themselves.

Because every mathematical object can be viewed as a set, any mathematical statement can be written in set theory notation and therefore any mathematical theorem can be derived using ZFC set theory. The reason ZFC set theory is so well known among mathematicians is, because of this, it is at the foundation of almost all modern mathematics.

http://www.britannica.com/EBchecked/topic/625445/John-Venn

http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory

http://plato.stanford.edu/entries/set-theory/#SetTheFouMat

http://www.encyclopediaofmath.org/index.php/ZFC

http://www-history.mcs.st-and.ac.uk/HistTopics/Beginnings_of_set_theory.html

http://en.wikipedia.org/wiki/Venn_diagram

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s