Tag Archives: set theory

Setting the Table for Set Theory

To understand set theory is to know its importance to other mathematical objects as well as understanding set theory itself. I will impart to you what I know about set theory and show you conceptually why set theory is so important.

Let’s start with the foundations first and work our way up. The basic operations of sets include unions, intersections, and subtraction. Complements, universal set, and strict subset are some other important basics that will be covered.

A union of a set is essentially a combining of the two sets we write it A⋃B. The only tricky part about a union of sets is to make sure you don’t over count. Basically, every object must be a unique mathematical object. To show this let A equal the set whose elements are three, four, and five. Let C be the set whose elements are three, six, and eight. The union of these two sets would be three, four, five, six, and eight. Notice how I didn’t say the number three twice because each element in a set must be unique (Khanacademy.org).

Intersection of two sets denoted A⋂B, involves taking the elements or element that both a set has in common with another set. If you had a set Z that had Je, Mouche, and Une Femme and another set Q that had Mouche, Fille, and L’ours. The intersection of these two sets would be Mouche because it is the only element that is common to both sets. A lot of mathematicians think of the intersection operator as “and,” for me it is also helpful to say “and” when reading the intersection symbol (Khanacademy.org).

Intersection (1)A physical example of intersection would be black and white. Where the black and white colors have combined making gray is the intersection, and everything else (the white and black colors remaining) is ignored—see figure one. The ⋃ symbol is used for unions and you can think of this as your body being separated at the waist. You have an upper portion with your arms, hands, etc. and a lower portion with your legs, feet, etc. The union of these two separated parts is your body.

Subtraction is not commutative amongst sets and is not too difficult to understand. The idea of subtraction amongst sets is defined as taking elements out of the first set that are in common with the second set leaving you with a new altered set that once was the first set as the remaining set. Say that you are a greedy capitalist whose name is Apu and you want to cut your unprofitable Kwik-E-Marts. You make a list/set that contains all of your stores and another list/set that contains all of your unprofitable stores. You then subtract the unprofitable list from the list of all your stores leaving you with a new set of just profitable stores. What this example doesn’t demonstrate is the fact that if the sets don’t have any elements in common, then the set of profitable stores won’t be changed.

The universal, complement, and strict subset are very closely related. The universal set is everything in the universe or rather everything in the universe that you might want to be concerned with. The universal set leads to what is called Russell’s Paradox. The paradox comes about from specifying the properties of what it means to be a universal set—the properties make it impossible to define what a universal set is without a contradiction occurring (Wikipedia). The complement of any set is everything not in that set. A strict subset has some of the elements of the set that it is a subset to, but not all of the elements (Khanacademy.org).

Notation basics are the arrow and the superscript c. The arrow looks very much like what you would see in your first semester class of Calculus (→), but it means that a particular domain has a certain codomain. An example of the arrow notation is x → y, which means x maps to y or the domain of x has the codomain of y. Just in case anyone is not familiar with domains and codomains they come from functions. A function is a particular type of relation that has only one output for a given input. The interval of possible inputs is known as the domain and the interval of possible outputs is the codomain. The superscript symbol AC, is used to denote the complement of a set. A complement isn’t the opposite of an object but the other part that makes a whole. Using the waist analogy again you can say that the complement of your lower half is your upper half and vice versa.

The importance of set theory comes from its very abstract nature. Strictly speaking a set is a collection of elements that have no particular structure other than the fact that you can verify that none of the elements are equal to each other and you can count the number of elements. Counting the number of elements in a set is known as the cardinality. The high level of abstraction in set theory allows you to represent all kinds of different math. Being able to translate other types of math into a single subject gives humans the ability to see how the other types support each other, are related, or if there are no connections (Lawvere and Rosebrugh). A simple example of this is how you can compare manifolds to Cartesian coordinates which I will explain further in the next paragraph.

The significance of set theory can be seen with the example that manifolds can be concisely described by a set with certain properties. Manifolds are mathematical objects in the area of topology that mirror Euclidean space near every point on the manifold. The power of manifolds lies in their complexity because they allow us to model complicated situations (Wikipedia). A specific way that you can represent a manifold is by using set theory to create neighborhoods. Neighborhoods essentially are sets that have three specific properties that you can use to say that points are near each other on a manifold. You can map the points of the neighborhoods onto a Cartesian plane and then perform transformations or inverse operations. This makes it easier to interact with these complex objects.

You are now a novice set theorist and you have a vague notion of why set theory is so important.

Works Citied

Lawvere, F. W., and Robert Rosebrugh. Sets for Mathematics. Cambridge, UK: CambridgeUP, Print.

“Manifold.” Wikipedia. Wikimedia Foundation. Web. 19 Mar. 2015 <http://en.wikipedia.org/wiki/Manifold&gt;.

“Khan Academy.” Khan Academy. Web. 2 Mar. 2015. <http://www.khanacademy.org/&gt;.

“Russell’s Paradox.” Wikipedia. Wikimedia Foundation. Web. 10 Mar. 2015 <http://en.wikipedia.org/wiki/Russell’s_paradox&gt;.

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.