Sets

 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 property.

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. You simply list each element, separated by a comma, and then put some curly brackets around the whole thing.
This is the notation for the two previous examples:
Notice how the first example has the "..." (three dots together).
So that means the first example continues on ... for infinity.
(OK, there isn't really an infinite amount of things you could wear, but I'm not entirely sure about that! After an hour of thinking of different things, I'm still not sure. So let's just say it is infinite for this example.)
So:

• The first set {socks, shoes, watches, shirts, ...} we call an infinite set,
• the second set {index, middle, ring, pinky} we call a finite set.

But sometimes the "..." can be used in the middle to save writing long lists:

Numerical Sets

So what does this have to do with mathematics? When we define a set, all we have to specify is a common characteristic. Who says we can't do so with numbers?
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 property, they are just defined that way. For example:
Are all sets that I just randomly banged on my keyboard to produce.

Why are Sets Important?

Sets are the fundamental property of mathematics. 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 mathematics that they are.

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

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 .

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

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:
So let's use this definition in some examples.
Let's try a harder example.

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 in A an element in A? (Yes, I wrote that correctly.)
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.
You should notice that if A is a proper subset of B, then it is also 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.

Empty (or Null) Set

This is probably the weirdest thing about sets.
As an example, think of the set of piano keys on a guitar.
And right you are. It is a set with no elements.
This is known as the Empty Set (or Null Set).There aren't any elements in it. Not one. Zero.
It is represented by
Or by {} (a set with no elements)
Some other examples of the empty set are the set of countries south of the south pole.
So what's so weird about the empty set? Well, that part comes next.

Empty 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 empty set a subset of A?
Going back to our definition of subsets, if every element in the empty set is also in A, then the empty 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 empty set that aren't in A, so it must be that all elements in the empty set are in A.
So the answer to the posed question is a resounding yes.

Order

No, not the order of the elements. In sets it does not matter what order the elements are in.
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.
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.

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