Now, in order to figure out an itinerary, I can't say figure out the itinerary for this function.
I need to give you a starting value.
So, I could do something like this:
Here's a seed. Here's a function,
and I could say: Find the orbit,
and that would be a pretty straightforward task.
You might need a calculator to evaluate these numbers,
but you just do the rule over and over again.
This function tells you how to find the orbit.
So, if I gave you this task:
Here's a function. Here's a seed. Find the orbit.
You would be certain that that task is possible.
There's no real way that that could fail, or that the question could be ill-posed,
and so in mathematics we would say that the solution,
in this case, I mean a solution to this question: Find the itinerary.
I would say that the solution exists.
So, let me write that.
So, if the problem I gave you was to find the itinerary,
given this and this,
we know that the solution exists.
There is no ambiguity. There's no way this process can fail.
So, the solution is guaranteed to exist.
Moreover, there's only one possible solution to this.
If I find the itinerary 2 , 4, 5, and then it keeps going, I can keep applying this.
Anybody else will find the same itinerary.
Darren, down the hall, would find the same itinerary;
Oscar in the Dominican Republic; Cedric in France;
all of us will find the same itinerary.
Another way of saying that, is, if somebody else finds an itinerary,
I don't need to look for any others,
because I know there's only one possible answer to this.
So, in math we would say that the solution is unique.
So, we would say that the solution is unique,
and these two statements taken together are equivalent to the following:
There is one and only one solution to this problem.
The problem here being find the orbit of this iterated function.
I hope, for this iterated function example,
this statement seems obvious enough to be almost pointless,
but, when we look at differential equations,
the existence and uniqueness might be a little bit more subtle,
so, I think it's good to see the notion of existence and uniqueness
first in this simpler setting.