00:01
We're given three sets, and we want to use two of these sets to make a couple of functions.
00:05
So our first function is going to be one that is both one to one and onto.
00:10
And we'll call that first one, f.
00:13
So in order for it to be one to one and onto, the fancy word for that is a bijection.
00:19
One thing that's necessary is the sets have to be of the same size.
00:25
So when we set up the function, it's going to have to relate x and z.
00:30
So we can just say f connects x to z and then we can say that it's defined by this set of ordered pairs now which element of x goes to which element in z is irrelevant but in order to be one to one and onto that means every element has to go to one and every element has to come from one so the most straightforward one would be a goes to s b goes to t and then a c goes to you but you can have any rearrangement of those as long as no input value appears more than once, or we should say every input appears exactly once, and every output also appears exactly once.
01:08
So that's one for our first set.
01:11
Now for our second function here, we'll say it's g.
01:14
We want it to be onto, but not one to one.
01:18
So when we want an onto function, that means the easiest way to do that with a finite set is to have the domain be larger than the co -domain.
01:27
So let's let g go from, y to say z since y has more elements in it than z does it cannot be one to one because remember one to one means every input value goes to only one output or goes to a different output we should say so for example let's look at this and again we can set it up however we want one can go to s in fact as long as every element goes somewhere, then it's going to be forced to be not one -to -one.
02:02
If we want it onto, everything in the code domain has to be covered.
02:07
So everything in z has to be covered.
02:11
So we can say three and you.
02:15
So here's a function that looks a lot like the original one...