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).
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.
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>.
“Khan Academy.” Khan Academy. Web. 2 Mar. 2015. <http://www.khanacademy.org/>.
“Russell’s Paradox.” Wikipedia. Wikimedia Foundation. Web. 10 Mar. 2015 <http://en.wikipedia.org/wiki/Russell’s_paradox>.