Definition 1.1.1. (A not so formal definition of sets).
A set is a collection of objects. The objects are referred to as elements or members of the set.
Section 1.1 Not so formal definition
In mathematics and elsewhere 6 we are used to dealing with collections of things. For example
- a family is a collection of relatives.
- hockey team is a collection of hockey players.
- shopping list is a collection of items we need to buy.
Let us give our first definition for the course. Now this one is not so formal — but it will be enough for our purposes 7 .
One reason to be not-so-formal here, is that while the notion of a set is relatively simple and intuitive, it turns out that making the definition completely rigorous is quite difficult. The interested reader should search-engine their way to discussions of this point.
Now — let’s just take a few moments to describe some conventions. There are many of these in mathematics. These are not firm mathematical rules, but rather they are much like traditions. It makes it much easier for people reading your work to understand what you are trying to say.
- Use capital letters to denote sets, \(A,B, C, X, Y\) etc.
- Use lower case letters to denote elements of the sets \(a,b,c,x,y\text{.}\)
So when you are writing a proof or just describing what you are doing then if you stick with these conventions people reading your work (including the person marking your exams) will know — “Oh \(A\) is that set they are talking about” and “\(a\) is an element of that set.”. On the other hand, if you use any old letter or symbol might be correct, but it can be unnecessarily confusing for the reader 8 . Think of it as being a bit like spelling — if you don’t spell words correctly people can usually understand what you mean, but it is much easier if you spell words the same way as everyone else 9 .
We will encounter more of these conventions as we go — another good one is
- The letters \(i,j,k,l,m,n\) usually denote integers.
- The letters \(x,y,z,w\) usually denote real numbers.
So — what can we do with a set? There is only thing we can ask of a set:
“Is this object in the set”
and the set will answer
“yes”
or
“no”
and nothing else. If you want to know more than just “yes” or “no”, then you need to use with more complicated mathematical structures (we’ll touch on some as we go along).
For example, if \(A\) is the set of even numbers we can ask “Is 4 in \(A\)” we get back the answer “yes”. We write this as
\begin{gather*}
4 \in A
\end{gather*}
While if we ask “Is \(3\) in \(A\text{?}\)” and we get back the answer “no”. Mathematically we would write this as
\begin{gather*}
3 \notin A
\end{gather*}
So this symbol “\(\in\)” is mathematical shorthand for “is an element of”, while the same symbol with a stroke through it “\(\notin\)” is shorthand for “is not an element of”. Similar “put a stroke through the symbol to indicate negation”-notation gets used a lot in different contexts and we’ll see it throughout this text. While it is arguably not terribly creative, it is effective — perhaps because it isn’t too creative.
This is standard notation — it is very important that you learn it and use it. Do not confuse the reader, or the person who marks your tests and exams, by using some variation of this. For instance, some of you may have previously used \(\varepsilon\) in place of \(\in\) — please stop doing so. For most mathematicians, “\(4\varepsilon A\)” denotes the product of three things, while “\(4 \in A\)” is a mathematical sentence that tells us that the object “4” is a member of the set \(A\text{.}\)
Subsection 1.1.1 Who is this reader you keep on mentioning?
We have referred to a “reader” several times in the text above but not really explained who we mean by “the reader”. There are 3 different types of reader that we mean when we say “think about the reader”: you, another person, and not-a-real-reader.
- You: Frequently, the only person who will read your mathematics is you. Your lecture notes, your homework drafts, your experimenting, etc — you typically don’t show them to other people. For that sort of work in isolation it doesn’t really matter too much if you don’t use standard notation, take shortcuts, and a myriad of other things that people typically do to save time. However, if we only think of ourselves when we write then we can form many bad habits that we take with us when we write for other people. These shortcuts can be hard for other people to understand unless we take the time to explain them. Consequently, it is a good idea to avoid these habits even when writing for ourselves; your reader, and even your future self, will thank you.
- Another person: On many occasions another person will read your work — the most obvious being the person who marks your homework, tests and exams. Generally you will not be present while they read (and perhaps grade) your mathematics, so typically they will only be able to mark what you have written on the page; they cannot mark what you mean by what is on the page. So you need to make sure things are as clear as possible, so that what you have written conveys what you mean. If you are in the habit of using your own shorthand or definitions or notation, then you must make sure these are clearly explained.
- Not-a-real-reader: Finally, we should often think of a reader who isn’t really a reader at all, but really just a mechanism we should use to decide if what we are writing is good enough. Our imaginary reader is intelligent, sensible, knows some mathematics (but not everything), and is a bit of an annoying pedant 10 . As we write we should think of this imaginary reader looking over our shoulder asking questions like “Is that the right notation?”, “Is that clear enough?”, “Does the logic flow in the right direction?” and offering advice like “Add another sentence to the explanation.” and “Make sure you define that function.”
As we continue along in this text we will keep referring to these readers and reminding you to think of them as you write. Communicating mathematics is a very important part of doing mathematics.
Including the so-called “real life” that non-mathematicians inhabit.
Unfortunately the formal theory of sets gets very difficult very quickly and is well beyond the scope of this text. So rather than investing a large amount of time on the precise definition of set, we will make do with this one. It is better for us to just get on with learning how to give precise definitions of particular sets and how to work with them.
While obfuscation can be useful in many endeavours, the authors do not know of any good reason to deliberately obfuscate your mathematics.
Okay, maybe Noah Webster had some not completely unreasonable reasons for tweaking English spelling, but this author is not entirely convinced that quite so many z’s are needed.
Is there any other sort of pedant?
