Okay.

While I'm still making the slides again, are there any questions?

We've been talking about proofs and so on

but we've recently kind of graduated to...

This is too loud.

We've graduated to actually using mathematical language in a particular and important area

which is discrete math.

Discrete math is all about sets and functions and relationships and those kind of things.

And where are we?

These are the slides.

We've started with something that we call naive set theory.

Naive set theory because as we saw

this set theory

this way of talking about sets while being the foundation of mathematics

has a problem.

Namely

if you start thinking about the set of all sets that don't contain themselves

we land in trouble.

So certain really, really, really big sets get us into trouble.

We are going to respond to that by just not looking at the really

really

really big sets here.

Okay, so we can actually stay naive.

If we wanted to do it safe

then we would have to do a lot more work.

We can make it safe

but you don't want to do that.

Okay

and mostly kind of what I'm doing

I mean

stay naive and kind of know about a problem somewhere out there is what mathematicians do.

Unless you really go into big sets

you don't really have to worry about things.

And we talked about sets

essentially how we make them by putting curly braces around stuff.

How if we have sets

how we can compare them

What's the concept of a subset, a proper subset, all of those things?

And operations on sets

how you

if you have sets

you can make other sets.

And we talked about sizes of sets only to realize that we have a problem.

Okay, so we have to get back to this when we have a little bit more machinery.

And the machinery is going to be functions.

So, the function is something like mapping objects to numbers.

And this understanding

this process of mapping things to other things is something I want to go into next.

And we're going to do that via the notion of a relation.

So what's a relation?