 Let's do a little bit more about mappings, and I just want to cement one of the issues for you And that is just the difference between a domain and a range. So let's have the domain Let's have the domain of the set of mine a1 Let's that that be the set of all natural numbers And I'm going to use a mapping that maps the natural numbers set of natural numbers to the set of natural numbers that is going to be my mapping and Here's a new thing I'm going to give my mapping a name and my mapping is going to be called alpha for instance alpha maps in to n plus 2 Such that in is an element of the natural numbers. So this is my mapping I'm mapping from the natural numbers to the natural numbers Because whatever natural number I put in there if I had to do that I'm still within the set of natural numbers later on. We'll see that has a specific meaning and And let's now just look quickly then again at the domain of a sub 1 if that's my natural number Now I'm mapping to this and I'm going to call this a sub 2 Now let's just look at the co-domain then the co Domain so that is would be this 2 that is also going to be the set of natural numbers But I'm mapping the natural numbers to the natural numbers by this mapping n goes to n plus 2 We know though that the elements 1 comma 2 is not in the Is never going to appear in this a sub 2 because if I start with 1 1 plus 2 is 3 So I'm going to 3 4 5 6 so this is never going to appear appear so that means my range For this mapping alpha is going to be the elements 3 4 5 6 Etc etc for infinity so that is really the difference then between a co-domain and Arrange the next thing I just want to if I just to have a quick look at if I have a sub 1 and I have a sub 2 and I have a sub 3 I have this mapping and I'm going to make it a B and C. I'm going to make it D and E I'm going to make it F and G and we're going to map There we're going to map there we're going to map there And we're going to map there Let's call my mapping from a 1 to a 2 let's call that alpha that maps the set a 1 to the set a 2 All we write now is something like this alpha of a equals D and the alpha of B Equals E and the alpha of C also equals E Can you remember what is this what would this be that indeed is onto because at least Each of the elements in the co-domain has at least one element in the domain mapping to it So that's how we would write there now imagine now we have beta beta that maps a 2 to a 3 That means the beta of D is going to be F and the beta of E is going to be G So what would this a 2 to a 3 be if you can think of that well There's at least one and every one is there So this is actually a bijection because it's both 1 to 1 and it's onto now I just want to talk to you about this Mapping that we have here. We're going to call this the beta alpha mapping. That's one way to write it Or I can say alpha Beta that's almost like a conjugate there But note that beta goes first in this because if I write the beta of the alpha of say for instance a Well That's that or that that's the same way to write it the alpha of a is D So here I'm looking at the beta of D and the beta of D was if In the beta of D was if so just a bit of This notation for you to get used to Now we're going to just I'm just going to mention the theorem. We're not going to prove it yet. That says if alpha If alpha is a bijection Then alpha has a unique inverse if alpha is a bijection Then alpha has a unique Inverse Now, what do I what I mean by that is that I must somehow be able to get back from my Codomain to my domain and that is going to be an inverse and that's why we said bijections are so important So I didn't say it yet. Don't think about this alpha. There's not the same alpha I'm just saying if a mapping is a bijection Then it has a unique inverse and the other quick thing just to have a look at if it is a bijection And then we're going to have something like this alpha beta's inverse is going to be beta inverse Alpha inverse don't want to say too much about that now We're going to have a look at the proof the important one of this if a mapping is a bijection Then it has a unique inverse So just to cement or just add a little bit of flesh to your understanding of Mappings before we get to do some exercises