Lesson 1a: Sets

Most of the stuff I'll be doing has to do with something called "sets."  The meaning is deceptively simple:

A set is an unordered list of elements.

Simple, right?  No problem at all. But what's an element?  Again, not as complicated as it sounds.

An element is anything that is in a set.

Sets are indicated by capital letters, like A, B, C.  I'd hope that you'd know what a capital letter is.  Elements can be anything at all.


Let's have an example.  Say that A is a set containing the numbers 0 through 20.  In order to write this, you'd say:

A = {0, 1, 2, ... , 19, 20}

The "..." is an ellipsis, indicating that the pattern of numbers continues, eventually coming to an end at the number 20.  The things inside the {} brackets are the elements of set A.

By itself, this set is just...there.  Nothing's happening to it.  Let's say that this was a secret code stored on a computer to make things more interesting.  A spy needs to find a number inside the set A.  He doesn't know what elements are in A, so he asks:

Is 20 an element of A?

This can also be written as:

Since the number 20 is indeed an element of the set A, the statement is true, and the spy gains access to the computer.  While that true is not important at the moment, it will be as we go on.

Now, let's suppose instead, the spy asked something that would have been false, like:

Is 600 an element of A?

We can see this is clearly not true, as A only contains 0-20.  So the computer would say back to the spy:

This can be described as:

600 is not an element of A.

There are also special types of sets that are denoted by weird letters.

These sets contain an infinite amount of numbers, so they can't be written out like our example set A could be.  These are also useful for describing a set.

There's also the empty set, which contains no elements.  That's denoted by the following symbol.

Despite this set being empty, it is actually important in describing something called set disjunction, which is beyond the scope of this post.

If you can understand all this, then pat yourself on the back, because you now understand basic sets!