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.
As you read the lectures, I recommend that you do a few things.
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!
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.
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.
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.
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:
{e, f, h, q} is how we would write out the set containing the letters e, f, h, and q. e, f, h, and q are all ELEMENTS of this set.
{δ, β, γ, λ} is how we would write out the set containing the Greek letters δ, β, γ, and λ. δ, β, γ, and λ are all ELEMENTS of this set. Notice that a set can contain anything; you don't need to understand what the things are inside the set in order to recognize that it is a set or to see how many elements are in the set.
{x −1, x2, 3x+4} is how we would write out the set containing the equations x −1, x2, and 3x+4. x −1, x2, and 3x+4 are all ELEMENTS of this set.
{red, yellow, blue} is how we would write out the set containing the words red, yellow, and blue. Red, yellow, and blue are all ELEMENTS of this set.
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:
{1,2,3,…,2000} is how we would write out the set containing all whole numbers from 1 to 2000, including 1 and 2000. The ellipsis, or three periods that come after the number 3, indicates that the numbers go on in between 3 and 2000. So this set contains not only 1,2,3, and 2000, but also 4, 5, 6, etc…, the numbers 1000, 1001, etc…., the numbers 1997, 1998, and 1999. So overall, this set has 2000 numbers as elements!
{0.5, 1, 1.5, 2, 2.5, ... , 57.5} is how we would write out all numbers from 0.5 to 57.5 which are either whole numbers or numbers with a decimal part which is equal to five tenths. So this set contains not only the numbers 0.5, 1, 1.5, 2, 2.5, and 57.5, but also 34.5, 10, 45, 57, and 22.5, for example. Overall, this set has 115 elements, although if you try to count them one by one, it might take a little time to count them all.
{c,d,e,…,r} is how we would write out the set containing all the letters of the English alphabet between c and r, including c and r. So while it is obvious that the letters c, d, e, and r are elements of the set, other element in the set are f, g, h, etc…., l, m, n, etc…., p, and q. So overall, this set has 16 letters as elements!
{1,2,3,…} is how we would write out the set of all positive whole numbers. The ellipsis, or three periods that come after the numbers, indicates that the numbers keep going on forever to the right. This means that not only are 1, 2, and 3 elements of this set, but also any positive whole number you can think of is in this set: so 4 is in this set, and 2984 is in this set and 2,038,847,382,394 is in this set. This set goes on forever, so it has in infinite number of elements in it. But also notice that, since it does not have an ellipsis to the left, the numbers do not go on forever on the left. 1 is the smallest element of this set. The number 0 is not in this set, nor is any negative number, because this set stops on the left at the number 1.
{…-3,-2,-1,0,1,2,3,…} is how we would write out the set of all whole numbers, both positive and negative. Notice that we have the ellipsis on both ends here, to indicate that the numbers keep on going forever in both directions. So some of the elements of this set include: -4, -3492, -23,468,044,958, 4, 5, 94, 264, and 666,994. This set has no smallest element and no biggest element; it just keeps on going forever both on the right and on the left. This set also has an infinite number of elements in it.
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.
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.
For example, Z is WELL-DEFINED because every number is either in the set of integers or not. We don’t have any numbers that could be both and integer and not an integer at the same time.
However, the set {x| x is a big number} is NOT WELL-DEFINED because we cannot tell how “big” a number must be in order to be an element of this set. Is 100 a “big” number? How about 100,000? Or 100,000,000,000? It is unclear how big a number must be to be considered “big.” In order for this set to be WELL-DEFINED, we need to be more specific. A WELL-DEFINED set that gets across the same idea might be the set {x| x > 1000}. This set is WELL-DEFINED because every number is either bigger than 1000 and therefore an element of the set, or smaller than or equal to 1000 and therefore not 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.
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:
2∈{x| x is a positive whole number}
This means “2 is an element of the set of positive whole numbers.”
8.9∉{x| x is a whole number}
This means “8.9 is NOT an element of the set of whole numbers.”
2∈{1,2,4}
This means “2 is an element of the set containing the numbers 1,2, and 4.
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:
The elements of this set are the positive and negative whole numbers, including 0.
We often use the symbol Z to represent the set of integers. Why do we use a Z instead of an I? Many of the mathematicians who worked in set theory were originally German, and Z stands for the German word Zahl, which means “number.”
Notice that the set of natural numbers, or N, is just the set of POSITIVE INTEGERS, which we can write as Z+. Also notice that the set of whole numbers is just the set of NON-NEGATIVE INTEGERS, or in other words, the set of all integers that are not negative (meaning all the positive integers and zero), which we can write as Z*. (Note: 0 is neither positive nor negative.)
This is a set that is basically impossible to write just by listing all the elements - what number would we start with? What is the smallest possible fraction? The biggest number which can be written as a fraction? What is the fraction which is closest to zero? Since we can't answer any of these questions, we can't easily write a list of all the possible fractions.
Instead we can write: 
, which we read as, “the set of all fractions p/q such that p and q are integers and q is not equal to zero.”
There are many numbers that can be written as fractions but which don’t always have to be in fraction form. A rational number is any number that can be put into fraction form, but which may not necessarily be written as a fraction. For example:
The number 8 is a rational number because it can be written as the fraction 
(or 
or 
, etc…).
The number 6.245 is rational because it can be written as the fraction 
.
The number 0.3 (this means the 3 repeats forever so we get: 0.33333333…..) is rational because it can be written as the fraction 
.
The number 0 is a rational number because it can be written as the fraction 0/1 (or 0/2 or 0/4623, etc...)
The word RATIONAL comes from the word “ratio,” which is just another fancy name for a fraction. (Notice that all integers and all decimals that terminate (end) or that reach a point at which the digits repeat forever in a regular pattern CAN be written as fractions. So 0, 2.6, and 3.1267 (the bar over the numbers means they repeat forever so we get: 3.126712671267… going on forever.) are all RATIONAL NUMBERS, but √2 and π, which are both decimals that go on forever without repeating, CANNOT be written as a fraction, and therefore are NOT RATIONAL NUMBERS.
We often use the symbol Q to represent the set of rational numbers. (Q stands for quotient. Quotient is just another fancy word for fractions or division. It is also the German word for fraction.)
These are all the numbers you can think of that CANNOT be written as a fraction, or, in other words, any decimal that goes on forever without repeating.
Some irrational numbers are √2, √3, √99, π, and e.
Irrational numbers are kind of an abstract idea. In the real world, we cannot actually calculate anything with an irrational number. For example, when you work with π, you are not actually using an irrational number in your calculations. You take the IRRATIONAL number π, round it off to the RATIONAL number 3.14, and then make your calculations. Irrational numbers are a very important idea, because they allow our number system to be CONTINUOUS. Because we have irrational numbers, we know that no matter what two numbers we pick, there will always be another number in between them. For example, to get a number in between 2.347285 and 2.347286, all we need to do is add another decimal place and we can get 2.3472855 or 2.3472851, etc. Because irrational numbers exist, we can keep adding extra decimal places on the end of a number forever. This actually means that between any two numbers, there is an INFINITE quantity of other numbers. If we didn't have irrational numbers, we wouldn't be able to depict numbers using a solid number line; instead we'd have to depict numbers using a sequence of disconnected dots with space in between them.
Unfortunately, there is no commonly used letter or symbol to represent the irrational numbers.