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Introduction to Sets

Forget everything you know about numbers. In fact, forget you even know what a number is. This is where math starts. Instead of math with numbers, we will now think about math with "things".

Definition

What is a set? Well, simply put, it's a collection. First you specify a common property among "things" (this word will be defined later) and then you gather up all the "things" that have this common trait.

For example, the items you wear: these would include shoes, socks, hat, shirt, pants, and so on.

I'm sure you could come up with at least a hundred.

This is known as a set.

Or another example would be types of fingers.

This set would include index, middle, ring, and pinky.

So it is just things grouped together with a certain property in common.

Notation

There is a fairly simple notation for sets. The two previous examples:

{socks, shoes, watches, shirts, ...}
{index, middle, ring, pinky}

Notice how one has the "...". All this means is that the set continues on for infinity. There may not be an infinite amount of things you wear, but I'm not entirely sure about that. After an hour of thinking of different things, I'm still not sure. The first set we call an infinite set, the second set we call a finite set.

Numerical Sets

So what does this have to do with math? When we define a set, all we have to specify is a common characteristic. Who says we can't do so with numbers?

Set of even numbers: {..., -4, -2, 0, 2, 4, ...}
Set of odd numbers: {..., -3, -1, 1, 3, ...}
Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}
Positive multiples of 3 that are less than 10: {3, 6, 9}

And the list goes on. We can come up with all different types of sets.

There can also be sets of numbers that have no common trait, they are just defined that way. For example:

{2, 3, 6, 828, 3839, 8827}
{4, 5, 6, 10, 21}
{2, 949, 48282, 42882959, 119484203}

Are all sets that I just randomly banged on my keyboard to produce.

Why are Sets Important?

Sets are the fundamental property of math. Now as a word of warning, sets, by themselves, seem pretty pointless. But it's only when you apply sets in different situations do they become the powerful building block of maths that they are.

Math can get amazingly complicated quite fast. Graph Theory, Abstract Algebra, Real Analysis, Complex Analysis, Linear Algebra, Number Theory, and the list goes on. But there is one thing that all of these share in common: Sets.

Universal Set

At the start we used the word "things" in quotes. We call this the universal set. It's a set that contains everything. Well, not exactly everything. Everything that is relevant to the problem you have.
So far, all I've been giving you in sets are integers. So the universal set for all of this discussion could be said to be integers. In fact, when doing Number Theory, this is almost always what the universal set is, as Number Theory is simply the study of integers.
However in Calculus (also known as real analysis), the universal set is almost always the real numbers. And in complex analysis, you guessed it, the universal set is the complex numbers.

Some More Notation

When talking about sets, it is fairly standard to use Capital Letters to represent the set, and lowercase letters to represent an element in that set.

So for example, A is a set, and a is an element in A. Same with B and b, and C and c.

Now you don't have to listen to the standard, you can use something like m to represent a set without breaking any mathematical laws (watch out, you can get π years in math jail for dividing by 0), but this notation is pretty nice and easy to follow, so why not?

Also, when we say an element a is in a set A, we use the symbol to show it.
And if something is not in a set use .

Example: Set A is {1,2,3}. You can see that 1 A, but 5 A

Equality

Two sets are equal if they have precisely the same members. Now, at first glance they may not seem equal, you may have to examine them closely!

Example: Are A and B equal where:

  • A is the set whose members are the first four positive whole numbers
  • B = {4, 2, 1, 3}

Let's check. They both contain 1. They both contain 2. And 3, And 4. And we have checked every element of both sets, so: Yes, they are!

And the equals sign (=) is used to show equality, so you would write:

A = B

Subsets

When we define a set, if we take pieces of that set, we can form what is called a subset.

So for example, we have the set {1, 2, 3, 4, 5}. A subset of this is {1, 2, 3}. Another subset is {3, 4} or even another, {1}. However, {1, 6} is not a subset, since it contains an element (6) which is not in the parent set. In general:

A is a subset of B if and only if every element in A is in B.

So let's use this definition in some examples.

Is A a subset of B, where A = {1, 3, 4} and B = {1, 4, 3, 2}?

1 is in A, but 1 is in B as well. So far so good. 2 is in B, but 2 is not in A. But remember, that doesn't matter, we only look at the elements in A. 3 is in A but 3 is also in B. One more to check. 4 is in A, and 4 is in B. That's all the elements of A, and every single one is in B, so we're done.

Let's try a harder example.

Let A be all multiples of 4 and B be all multiples of 2. Is A a subset of B? And is B a subset of A?

Well, we can't check every element in these sets, because they have an infinite number of elements. So we need to get an idea of what the elements look like in each, and then compare them.

To represent a multiple of 2, we use 2n, where n is an integer. And then we do the same with the multiple of 4, 4m, where m is an integer. So if we have a number 4m, can we write it as multiples of 2, such as 2n? Of course we can!

We know that 4 = 2*2, so 4m = 2*2m or rather, 2(2m). We also know that 2m must be an integer. So let a = 2m, where a is an integer. Then we can say that 4m = 2*2m = 2(2m) = 2(a). Since a is just some integer, it's pretty much the same as 2n. I mean, all were doing is using a different letter, and the letter that you use doesn't matter. So A is a subset of B.

But is B a subset of A? Well, we can try the same thing. We have 2n and we want to make it look like 4m. One way that we could do that is to multiply by 2 so that we get 2*2n or rather 4n. But remember above, we were only allowed to use equal signs. If you multiply a number by two, you can't use the equals sign. So we have just ran into a brick wall. We can't get 2n to look like 4m. Maybe the statement is false? So instead, let's try to show that B is not a subset of A. How can we do that? Well, if we have only 1 member in B that isn't in A, we know that B can't be a subset of A. So all we have to do is find that element. We want a multiple of 2 that is not a multiple of 4. And in fact, 2 is a multiple of 2, but it's not a multiple of 4. So we found it, and thus, B is not a subset of A.

Proper Subsets

If we look at the defintion of subsets and let our mind wander a bit, we come to a weird conclusion. Let A be a set. Is every element a in A also an element in A? Well, umm, yeah, right? So wouldn't that mean that A is a subset of A? This doesn't seem very proper, does it? We want our subsets to be proper. So we introduce (what else but) proper subsets.

A is a proper subset of B if and only if every element in A is also in B, and there exists at least one element in B that is not in A.

This little piece at the end is only there to make sure that A is not a proper subset of itself. Otherwise, a proper subset is exactly the same as a normal subset.

So for example, {1, 2, 3} is a subset of {1, 2, 3}, but is not a proper subset of {1, 2, 3}.

On the contrary, {1, 2, 3} is a proper subset of {1, 2, 3, 4} because the element 4 is not in the first set.

You should notice that if A is a proper subset of B, then it must also be that A is a subset of B.

Even More Notation

When we say that A is a subset of B, we write A B.

Or we can say that A is not a subset of B by A B ("A is not a subset of B")

When we talk about proper subsets, we take out the line underneath and so it becomes A B or if we want to say the opposite, A B.

Null (or Empty) Set

This is probably the weirdest thing about sets.

As an example, think of the set of piano keys on a guitar.

"But wait!" you say, "There are no piano keys on a guitar!"

And right you are. It is a set will no elements.

This is known as the null set.There aren't any elements in it. Not one. Zero.

It is represented by

Some other examples of the null set are the set of countries south of the south pole.

So what's so weird about the null set? Well, that part comes next.

Null Set and Subsets

So let's go back to our definition of subsets. We have a set A. We won't define it any more than that, it could be any set. Is the null set a subset of A?

Going back to our definition of subsets, if every element in the null set is also in A, then the null set is a subset of A. But what if we have no elements?

It takes an introduction to logic to understand this, but this statement is one that is "vacuously" or "trivially" true. A good way to think about it is: we can't find any elements in the null set that aren't in A, so it must be that all elements in the null set are in A.

So the answer to the posed question is a resounding yes.

The null set is a subset of every set, including the null set itself.

Order

Every set also has an attribute to it known as order. It is simply the size of the set.

Just as there are finite and infinite sets, each has finite and infinite order. For finite sets, we represent the order by a number, the number of elements. So for example, {1, 2, 3, 4} has an order of 4. For infinite sets, all we can say is that the order is infinite. Oddly enough, we can say with sets that some infinities are larger than others, but this is a more advanced topic in sets.

Arg! Not more notation!

Nah, just kidding. No more notation.

Exercises

coming soon ...

by
Ricky Shadrach

 

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