Okay, so the quiz has finished.

There's still one quiz, I think quiz two, that needs a re-correction.

We haven't forgotten that.

There was one problem with the quiz.

So that's coming up still.

We're still using that as a guinea pig for the new re-correction mechanism.

Right, so are there any questions so far?

What we've been doing is we've been talking rigorously about sets.

And there's two reasons we're doing this.

One is it's important for you to learn rigorous math language.

Some of you have only kind of been doing engineering math,

which is just much more computation oriented,

but not actually language oriented.

Language as is important for mathematical proofs and definitions.

Those are the ones that we are much more interested in

and much more dependent on in symbolic AI.

So that's something you need to train.

That's rigorous language.

And the other thing is sets are everywhere.

Indeed it is the case that sets are somewhere down there

and we've looked at set constructors.

How do we make sets?

Bigger sets out of smaller sets, smaller sets out of bigger sets.

Okay, how do we compare sets?

One is a subset of another.

Two sets are equal and so on.

Two sets are disjoint, meaning they don't share any elements.

Those are relations on set.

And the next thing we started doing was,

and that's one step up and you need to understand that,

we've been looking at relations in and of themselves.

We humans have lots of relations, right?

Being in love, being married, being enrolled,

all of those kind of things, right?

Whenever there's a relationship between you and something

or between two numbers, one number divides the other

or something like that.

And we need to have a vocabulary on this.

So the next thing we did was basically we kind of extended

what we could talk about by relations.

And what we're going to do today is functions and so on.

And eventually what happens is that all of math,

and by extension all of rigorous computer science,

can be based on sets, which is why I'm torturing you with them.

If you can't do sets, you will have a problem later on.

Okay, so that's one of the reasons.

And kind of computer science is really kind of there.

Functions are a very important thing.

Relations are a very important thing.

Applied relations we call database theory.