This evening

keywords jump up and down,

Okay. I'm beginning to notice a number of international students. Is there a preference

to do the course in English? That's one yes, more yeses. Yeah, of course. Some of you know

that. Okay, we can put it to a vote. Who would agree to having the course in English? That's

everybody except one, right? Is there anybody who doesn't speak German? Half a person. Okay.

All right. That sort of leaves it on the edge, right? Okay. One point to add to that. The

course material will remain in German because it currently is in German and I'm not maintaining

two versions of the material. So under that proviso, who would agree to having the course

in German? Nobody. Sorry. No, that was really wrong. So under that proviso, the material

being in German, who would agree to the lectures being in English? Okay. So that is now a sort

of semi-unanimous vote. Okay. Let's try this. It will carry the slight problem that while

the material is in German, I'm really off the material, right? So bear with me. As some

of you may have noticed, the sheet's been online since this morning. Okay. One thing

to notice is the hand in date, so that's two weeks from now. Currently, you're not able

to solve any of the exercises that are on the sheet, but I'm hoping to cover most of

the necessary material today and the rest on Monday. So two weeks from now is like 11

days from Monday, which I guess should be enough time to solve it because it's also

not excessively hard. I'll show it to you now, although you don't yet know some of the

keywords just to basically raise flags when the words that are on the sheet actually crop

up in the lecture. So there's an exercise on currying and number one. Well, I mean,

many of you who knows about functional programming will of course know this, but we're nevertheless

going to make it explicit one more time. Then there's a number of exercises and number two

on just a particular format of recursive definitions that we're hopefully going to see either today

or on Monday, specifically the fold format. Then there's one example of an inductive proof

for functions that are defined in terms of fold will be defined in terms of fold in the

lecture. So defining them. So this basically, this is the map function of this, we just

call it list instead of map. So this will be seen in the lectures, defining it as not

part of the exercise. And then there's an exercise at the end that is slightly more

substantial where you're asked to actually define a data type of trees of a certain shape,

then come up with the, well, two laws of the general nature that we're going to see in

the lecture, as I said, even now on Monday. And to use the, well, generic induction principle,

if you will, that you get from this data type to prove a very simple equation. Okay. I'm

going to switch this off.

Sure.

Okay.

So the first session last week was a basic recap of things to know about sets and functions.

And well, we're going to delve a bit deeper into certain functions that are going to be

important in the course, and which in particular we're going to want to generalize NATO to,

well, when we think about categories other than sets.

So basically we're going to look at some important functions or actually types of functions.

So one operator on functions that we talked about last time is composition of functions,

which we wrote in applicative form, so f composed with g means execute g first then f.

And we have this notation for the function space x arrow y denotes the set of all functions

from a set x into another set y.

And first thing to note is that under this notation we can write the composition operator

itself as a map.

So we can call this comp to be definite.

We'll occasionally use it.

And well, it takes two arguments.