Set Theory

Set Theory involves the study of collections of things rather than the study of individual things. Often we are looking to describe a collection of numbers, or a collection of equations, etc. We may want to talk about their properties, or describe their relationship to another collection of things. You have, in fact, probably already encountered the use of sets in your math classes before - you just didn't know it at the time. First we will give some basic definitions so that we have a basic vocabulary to build on, and then we will come back to this idea.

How to read these lectures:

As you read the lectures, I recommend that you do a few things.

  1. Write down each new definition you encounter somewhere on a single sheet of paper to keep with you while you do problems, and make sure that you understand it exactly. Put the definition into your own words, but be very careful to make sure that your words are precise and correctly describe the definition. Getting questions right or wrong often depends upon being very specific and exact when you use definitions.
    Write down whatever you need to remember exactly what a new definition means - maybe give yourself some key examples of a given definition. You will not typically be quizzed on definitions in this class, but if you do not understand the major definitions in this class, you will have trouble understanding the lectures!

  2. For each example in the lectures, try to explain it yourself first and practice working it out on your own, before you look at the explanation given in the lectures. This is a good way to test your own understanding of the concept before you move on to the next section.

  3. If you find that you don't fully understand a concept after having tried a few problems on your own, try reading the lecture again. There is a lot of information in these lectures. You may actually find it useful to read a lecture through quickly first to get the main ideas, and then to go back and read it more slowing a second and third time to focus on the specific details. To really understand the math in this class, you will probably need to read each of the lectures more than once.

  4. Read slowly and carefully - if you don't understand a sentence the first time you read it, read it again slowly and try to break it down. You cannot read mathematics the way that you read a newspaper or a novel! No one, not even very successful mathematicians, can read about totally new mathematics without reading very slowly, rereading key sections several times, and writing down important concepts and definitions as they go. When I read a math article about new ideas that I don't yet know, I always write all over the article, circling any words that I don't know and want to look up somewhere else, working out any examples on my own to make sure that I really understand the concept, and writing down questions about specific parts of the article that I don't fully understand.
    If you read through a math lecture at the same speed that you normally read novels or magazines and you don't understand it, this is normal! Don't give up or make assumptions about your mathematical ability at this point - instead try reading the lecture again more slowly - you may even try printing out the lectures so that you can write directly onto them; working out the examples in the margins and writing down questions you have about the math as you go.

 

 

In mathematics, a SET is just a collection of things that have been grouped together, just as in regular English, like a “set of steak knives” or a “set of candlesticks.”

 

Since this is a math class, we will often be talking about a SET of numbers. But we might also talk about a set of variables, or equations, or points, or other kinds of mathematical objects. There is actually no limit to what kinds of things a set can contain.

 

Writing Sets:

We can write out a SET by just listing the ELEMENTS in the set in between curly brackets, each separated by a comma:

An ELEMENT of a given set is any thing which would appear in the list of things in the set. Each element in a set is separated from other elements in the set by commas. So, for example, the numbers 1, 2, and 3 are all elements of the set {1,2,3}, but only the number 123 is an element of the set {123}.

Be Careful! Notice that we use these squiggly brackets { } to denote a set. We cannot use these ( ) regular parentheses (or the square brackets [ ] ), because that means something else in mathematics.

For example, if you recall from a previous math class, (1,2) is an ordered pair which represents the point in the x-y-plane which is one space to the right and two spaces up. In an ordered pair, order matters, as we can see when we look, for example, at the ordered pair (2,1); this is the point in the x-y-plane which is two spaces to the right and one space up. So the points (1,2) and (2,1) are not the same, and neither (1,2) nor (2,1) should be confused with the set {1,2} or the set {2,1}. (If you are curious about what [1,2] means, that is used to denote the interval of all real numbers between 1 and 2, including 1 and 2. It is the same thing as writing 1 ≤ x ≤ 2.)

So be sure that when you write sets, you always use the correct kind of brackets: { }.

Here are some more examples of sets that have been written by just listing all of the elements in the set:

 

Writing Very Large Sets:

Sometimes, a set is really big, or even infinite; when this happens, it is really hard or even impossible to list every single element in the set. But as long as we have a set where the elements seem to come in some kind of logical order or pattern, we can get around this by using the ellipsis sign, which looks like 3 periods: “…”. If we have a really big set, we just need to list enough of the elements of the set to establish what the pattern is (this usually requires about 4 or 5 elements) and then we put in the ellipsis to indicate that the set has other elements that continue in a similar pattern in between the listed numbers.

 

For example:

 

What Order Should We Use When Listing the Elements of a Set?

You may have noticed that when writing sets, if the set contains numbers, we usually list the numbers from smallest to largest. This is just a convention that makes sets easier to read, but it is not actually necessary. For example, I could write the set {1,2,3,4} like this: {4,2,1,3}. When writing sets, the order of the elements does not matter. So, for example, {2,6,24}={2,24,6}={6,2,24}={6,24,2}={24,2,6}={24,6,2}.

The whole idea of sets is that we only care whether or not something is actually in the set; we don't care where it appears in the set, or how many times it appears. Listing an element more than once in a set also doesn't change the set; so, for example, the set {2,2,3} is exactly the same set as {3,2}, because both sets contain exactly the same elements. The first set contains only the numbers 2 and 3, and the second set contains only the numbers 2 and 3; the order of these elements or the number of times any element appears in the set is irrelevant.

We say that two sets are EQUAL if every element of the first set is an element of the second set and vice versa. So {f,i}={i,f,f,f,i} because:

So, these two sets are equal, because they contain exactly the same elements.

Writing a Set by Describing its Properties:

We often want to talk about a collection of things that have certain properties in common. The study of set theory was created because mathematicians wanted to develop tools to talk about whole collections of things. Often we are interested in discussing the properties of a certain set of numbers or of equations, and if we have a systematic way of describing these sets, a clear way of talking about relationships between sets and describing how one set can be created from a combination of one or more other sets, then we will be able to make a better study of them.

In fact, some sets would be really difficult to write out by trying to list all the elements, or even by trying to list part of the elements. For example, if I wanted a set to contain all the real numbers between 1 and 2, I wouldn't even know which number should come first in the set. What is the smallest number between 1 and 2? 1.1? 1.01?1.0000000000000001? No matter what number we pick, there will always be an even smaller number. So we cannot really write out this set by trying to list all or part of its elements. In fact, the clearest way to describe this set is simply to describe it in words: it is the set of all real numbers greater than 1 and less than 2.

So, rather than listing all or part of the elements of a set, we can also write out a SET by describing it in words:

Be careful when writing a set by describing its properties - whenever you do this, you must be sure that your set is still WELL-DEFINED. This means that we have to define our set clearly enough that we can always tell whether or not a number is an ELEMENT of the set.

We can always write a set either by listing all of its elements (or enough of its elements with an ... to give the pattern) or by describing the properties of the elements of the set; which method you prefer will often depend on how easy it is to list all the elements of a set. The first method gives us a clear view of everything that is an element of the set. The second method makes it really clear what properties the members of the set share.

Using Symbols to Denote the Elements of a Set:

Once we have a set, we often want to talk about what ELEMENTS are in the particular set we are looking at. We have a symbol that makes this easier to write:

 

Using Uppercase Letters to Represent Sets:

Sometimes we represent a set with a letter. For example, if we are working with the set {a,b,c,…,z}, we may just call it A, so that we don’t have to waste a lot of time writing out the whole set every time we want to refer to it. We would write A={a,b,c,…,z}, and then from then on we could just write A whenever we want to indicate this set of all lowercase letters of the alphabet. We use uppercase letters to denote sets. So notice that in this case a∈A (we do not have a=A!).

 

 

There are several SETS that you are already familiar with from basic math. You have already worked with sets; you just didn’t call them sets at the time: