Set Wars: Episode I - The Phantom Mathematics

Much of the early material in an introductory analysis class includes sets and their properties. Finite sets, infinite sets, empty sets. There are many elementary operations on sets such as union, intersection, complement and relative complement. Before we talk about these operations we must first define what a set is.

Definition: A set is a collection of mathematical objects.

This is a somewhat ambiguous definition. What are mathematical objects? Mathematical objects can be numbers, functions, other sets, you name it. Sets are typically written in the following way:

$ S = \{1, 2, 3, 4 \} $. (Meaning our set $S$ has for elements the numbers $1$, $2$, $3$ and $4$.)

Important examples of sets are the integers (as we mentioned before), rational numbers, irrational numbers, the real numbers, the natural numbers, the empty set, the list goes on and on. Now that we understand what a set is, we can talk about elements of the set.

You might wonder why we are concerned with sets and how on earth they could be important. For starters, sets give us a way of collecting objects in a meaningful way. If we are interested in the properties of some collection of numbers, it might be best to group all of them together in a set and prove general properties the collection has. Example: whether or not there are an infinite amount of prime numbers. There are at least two ways to attack this problem and both would be difficult without having some concept of a set - whether or not you use the word "set." Or perhaps you want to know if there are an infinite amount of irrational numbers. First you'd have to have a collection of them (read: a set of them) and then you'd have to devise a way of finding out if there are finitely many of them.

Concept: An object $x$ is said to be an element of a set $S$ if it is contained in $S$.

From our previous example of a set, the numbers $1$, $2$, $3$ and $4$ are all elements of $S$. If $x$ is an element of $S$, we write $x\in S$ as shorthand (read: $x$ is in $S$). We can now talk about subsets (a very important concept). The technical definition of a subset is somewhat difficult to swallow at first, so I will first refer to our previous example. Let $A$ be the following set:

$$ A = \{1, 2 \}.$$

$A$ is a subset of our previous set $S$ because the all of the numbers in $A$ are also in $S$. However, if $A$ included the number $9$, then $A$ would no longer be a subset of $S$ since it would no longer be contained in $S$ (i.e., it would have an element that is not in $S$). Now for the technical definition of a subset.

Definition: A set $A$ is said to be a subset of a set $B$ if and only if when $x$ is an element of $A$, then it is an element of $B$.

It may appear as though this definition isn't correct at first glance since we seemingly only mentioned one element - $x$ - in $A$. When stating definitions, mathematicians try to be as succinct as possible. The element $x$ is supposed to represent an arbitrary element in $A$, not a specific element and implicit in the statement is that we range over $x$. An alternative definition is as follows: A set $A$ is said to be a subset of $B$ if and only if all elements of $A$ are elements of $B.$

Minor definition: A set $A$ is a proper subset of $B$ if $A$ is a subset of $B$ and $A$ is not equal to $B$.

Question: Let $A$ be a set. Is $A$ a subset of itself? Why or why not? I will leave this to you to determine.

Before I mentioned something called the empty set. The empty set is exactly that: empty. It has no elements. The empty set is a subset of every set, and this will be shown later. There are multiple proofs of this claim, but the proof I intend to employ will rely on operations on sets we have not yet defined.

Operations on Sets

There are multiple operations on sets. Notable examples are union, intersection, complement and symmetric difference (this last one is important in abstract algebra). Union will be fairly similar to the logical disjunction from propositional logic and intersection akin to logical conjunction. For those who know more than basic logic, symmetric difference is akin to exclusive or.

Again the operations are easiest to see with examples first. Let $A = \{1, 2, 3, 4, 5\}$ and $B = \{3, 4, 6\}.$

The union of the two sets is the combination of the two without double counting common elements. So the union of $A$ and $B$ as above is then the set $\{1, 2, 3, 4, 5, 6\}$ (notice we did not double count). We could have double counted, but by convention we do not since sets only see membership, not frequency of occurrence.

The intersection of $A$ and $B$ is the set of elements they have in common; in this case, it gives the set $\{3, 4\}$. $3$ and $4$ are in both sets and are the only elements common to both, so these are the elements in the intersection of the two sets.

The relative complement is the weird one out of the bunch. Unlike the previous two, $A$ relative complement $B$ is not the same as $B$ relative complement $A$ in general (in mathematical terms, relative complement does not commute). Colloquially, $A$ relative complement $B$ is the set of the elements in $A$ which are not in $B$. In our case, it would be $\{1, 2, 5\}$ and $B$ relative complement $A$ is $\{6\}$.

Notation: The union of $A$ and $B$ is denoted by $A\cup B$. The intersection of $A$ and $B$ is denoted by $A\cap B$. The relative complement of $B$ in $A$ is denoted by $A\setminus B$.

There is one final major concept we must explore: the universal set. The universal set is the set from which you are drawing elements. Common examples include the integers or the real numbers. We have a notion of relative complement and we also have a notion of an absolute complement.

Definition: Let $R$ be your universal set and $A$ be a subset of $R$. The (absolute) complement of $A$ is the set of all elements in $R$ that are not in $A$. This is often written as $R\setminus A$ or $\bar{A}$ or even $A^c$. The barred and $^c$ notations are particular to absolute complement and are not used in relative complements.

Question: Let $R$ be a set. What is $R\setminus R$?

Answer: We know that $R\setminus A$ is the set of all elements in $R$ that are not in $A$. If $A = R$, then our result will be the set of all of the elements in $R$ which are not in $R$. There is not a single element that is both in $R$ and not in $R$ (as this would be a contradiction), so we conclude that $R\setminus R$ has no elements, i.e. $R\setminus R$ is the empty set. This makes sense since if you take all of the elements of $R$ out of $R$ you're left with nothing!

For the sake of keeping future posts short, I will end this post here. The next post will detail methods of proof concerning sets and some oddities that surround sets. Many strange results arise from infinite sets. Including (but not limited to) the fact that the integers have the same cardinality (think: size) as the natural numbers and that the irrational numbers far outnumber the rational numbers even though there are an infinite amount of both.

The Importance of Mathematics

This post is a change of pace and the mathematics included will be minimal. Before continuing with the first of multiple posts on set theory, I would like to explain the importance of mathematics and why you should study it (i.e., why you should continue to follow my blog!).

Since the dawn of mathematics, it has been used as a tool for understanding the natural world. Newton developed differential calculus in order to explain motion of moving bodies, differential equations have been used to study motion, population growth, and decay rates of materials, partial differential equations were used to study heat conduction in metals, the way waves propagate in media and the study of quantum mechanics. Mathematics is everywhere: in the fan circling above your head, in the satellites orbiting the Earth providing you with GPS navigation, in the warming body that we call the Sun. Each of these three topics will be treated in a physics blog I intend to create later down the road.

When grinding through a mathematics class, it is easy to miss the forest despite all of the trees. Sometimes you may feel inadequate as you are learning a new topic. It is imperative that you remind yourself that the world's best men had difficulty developing many topics you learn as an undergraduate. Mathematics was not built in a day. Newton and Leibniz developed calculus in the 1600s, but at the time calculus was considered mysterious. Newton's fluxions - as they were called - became what we call differentials (extremely small quantities); even the great Bishop Berkeley called these fluxions "ghosts of departed quantities" and many others felt similarly. Calculus was heavily used by mathematical physicists (called natural philosophers at the time), but some true mathematicians considered calculus incomplete or possibly incorrect. It was not until the 1800s that Weierstrass gave us our current definition of what a limit is. As you can see, it was not an easy task even for the best mathematicians at the time to make calculus a rigorous topic in its own right. The moral of the story is that in your courses you are learning potentially hundreds of years of mathematics in a very short period of time, so do not be discouraged.

Having said that, mathematics becomes more abstract from here. Mathematics is wonderful because it has the ability to explain, but is not limited to, the natural world. Some people believe all of mathematics will find its way into explaining the world around us and it certainly is true that much of mathematics has. I am unsure of whether or not all branches of mathematics will find its way into explaining the natural world, but it is an exciting concept!

In the near future, I will make a post concerning abstractions found in mathematics. The idea of a factorial can be extended, the ideas surrounding vectors (as you probably used in physics in high school) can be generalized and so can the concept of a length. You might ask: why generalize these ideas? At first generalization can seem unwarranted or perhaps even unnecessary, but many times these "unnecessary" generalizations are used when modeling something in the real world. The abstraction of vectors into vector spaces and length is hugely important in quantum mechanics (as will be covered in my physics blog).

On that note, I end this post. The upcoming topic of set theory is very rich and full of intrigue and mystery, concreteness and abstraction. Some of the arguments may be difficult to internalize at first, but I will try to make them as clear and understandable as possible. And in case I don't see you, good afternoon, good evening and goodnight.