Topic 9: Sets

Definition

What is a set? Well, simply put, it's a collection.

First we specify a common property among "things" (this word will be defined later) and then we gather up all the "things" that have this common property.

For example;

The items you wear: 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 is types of fingers.

This set includes index, middle, ring, and pinky.

Notation

There is a fairly simple notation for sets. We simply list each element (or "member") separated by a comma, and then put some curly brackets around the whole thing:

The curly brackets { } are sometimes called "set brackets" or "braces".

This is the notation for the two previous examples:

{ socks, shoes, watches, shirts, ...}

{ index, middle, ring, pinky }

Notice how the first example has the "..." (three dots together).

The three dots ... are called an ellipsis, and mean "continue on".

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:

For example: The set of letters

{ a, b, c, ..., x, y, z }

In this case it is a finite set (there are only 26 letters, right?)

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?

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 property, 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 mathematics. Now as a word of warning, sets, by themselves, seem pretty pointless. But it's only when we apply sets in different situations do they become the powerful building block of mathematics 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.