You can hear me via the microphone

right?

Okay

so we were talking about sets

relations

which are sets of pairs

functions

which

are special relations

and we were using special functions

namely bijective

functions, for understanding sizes of sets, or cardinology as we say.

And so

there was an excursion about big sets like the real numbers that are uncountable.

Interesting.

Okay, so let me see.

Yeah, indeed, this is off.

So

until this is turned on

so we call a set countable if there's a bijection to the natural numbers

that are the smallest infinite sets

and there are sets that are bigger than the natural numbers

and I showed you this

I think

very nice diagonalisation argument.

Let me see.

Is it coming back?

Something's changing.

Maybe it's coming back.

Let's wait for a second.

That needs a while until it's back on.

There it's now. There's a chance it'll go back on.

So

the next thing I want to talk about is operations on functions.

Are there things that take functions and give us new functions

or that give us that...

Excellent.

Or operations that take a function and give us a set or something.

Right, so...

Yeah.

Now it would be nice if this went away.

Excellent. Good.

So, and there's one we've already seen.

We have talked about the composition of the functions.

So, let's see.

So, let's see.

So, let's see.

So, let's see.

So, let's see.

So, let's see.