Okay, quiz is over.

Let's restart.

Are there any questions so far?

So remember we're still trying to do two things at once, learn about sets and kind of learn

about talking about sets while we're doing it, learning about how to do math along the

way.

And last time I tried to tell you about ways of kind of doing multiple ways of doing things

in math and I kind of extemporized on that but I've since made slides and I want to basically

go over them very quickly.

The idea is that even though math kind of being the queen of all sciences or the language

of doing math and you would like to think that for everything there's one right way

of doing it and everything else is wrong and that's not the case.

On the other hand there's multiple ways of doing things and they're good for different

purposes.

And what's more it's often extremely useful to kind of switch over to some other way of

doing things because that makes other things easy and other things and allows you new perspectives

that give you new insights.

And that practice is extremely ingrained when we do math so much that experienced mathematicians

aren't even noticing it.

Good mathematicians may become bad teachers because they don't understand what's difficult

anymore.

But for you it's legitimately difficult that there are things that have multiple ways of

doing it.

So you have my sympathy but I can't help it.

It's just the way it is and it's good, it's the way it is for good reason.

I would like you to basically talk about this some more.

So for you that means you need to be able to deal with the fact that there are multiple

ways and you will have to have some kind of an overview of what the multiple ways are

because you want to not only when you're solving problems switch perspectives yourselves but

you also need to be able to kind of follow others on the perspective switches.

And depending on the level at which you do math or computer science those jumps may be

little baby jumps or may be huge jumps that may require years of study that you actually

are able to do them.

At least for the last latter part of jumps people tend to apologize first.

Good okay so you'll have to tolerate different ways of doing stuff and you will have to become

good at switching yourselves.

That is not limited to symbolic AI.

That's all over computer science because kind of the language of computer science comes

from math.

That's the reason we train you for multiple semesters in math in the bachelor's program

and then some more in the master's program.

Okay so I have prepared two examples.

One of them I'm going to very briefly go through.

You remember that we were talking about bijective functions.

Bijective functions being injective or one to one functions which basically means injective

functions don't bring things together that were different in the domain and surjective

which are the functions that reach everything and bijective is both and there's another

way of saying that.

Another way of saying that it has its own name it's called invertible.

A function is invertible if we can essentially undo what the function did by basically going