 Now what is actually a function? A function is a special type of relation which should meet two criteria. A function is a special type of relation which should meet how many criteria? Two criteria and what are they? The criteria number one is... The criteria number one is... Every element... These are called elements. Every element in the pre-image set should have an image. That's criteria number one. Write it down. Every element in pre-image set... Pre-image set must have... Image means image of one is four. So every element should have an image. Is that correct? Yes. There's no element in set A which is left unmapped. Then that relation would not be called as a function. Let's say there was another element five. Hypothetically, let's say there was a five. Correct? Or even five as in six. Let's say six. I think it looks like six one. Six. Six should have been mapped to nine. But is nine present in set B? No, so six cannot be mapped. So six is unmapped. And when six is unmapped... That means this criteria is getting violated. That means every pre-image does not have an image anymore. And then in that case, I would not call this relation as a function. So this is criteria number one. Criteria number two as what Sher's told is... The image must be unique for the same input. I should not get two answers for the same... For example, if I put one, in this case it will only give me four. It will not give me five or it will not give me ten. It will not give me hundred. Correct? Now you must be wondering when does that happen actually? When can I have two different outputs for the same input? For example, in this y squared is mapped to two also. It will map to minus two also. This should not happen in a function. So this will not be called a function. Are you getting it? Now people ask me, why are these two restrictions put on the relation to be called as a function? Plain and simple. Just try to relate it to functions in real life. What are functions? Is this mobile playing a function for me? Yes. I make calls from it. I see my messages and my views from it. Is the marker playing a role for me? Let's say simple toast machine. Toaster. Does it play a functionality for you? Yes. Now let's say you have a toaster in your house. Correct? And you try to put a piece of bread in it. And it doesn't give you any output for it. What will you do? Take it out of the house. Throw it out of the house. You say, what is this machine? It's not playing its functionality. Yes or no? So when you put something, it should give you an output as per the manual of the function. Isn't it? So if an input doesn't give you an output, then that particular mathematical machine is not functioning and hence not a function. Correct? Now again, the same machine, let's say, one day you put a bread and it gives you a toast. The same bread you put the next day and it gives you a pizza. I know you'll be happy to see that. But you say, oh, you're a Darba machine. One day it is giving me toast. The other day it is giving me pizza. Again, what will you do? Throw it out of your house, isn't it? So you cannot expect different outcomes from the same input. Bread means toast should come out. It should not give you surprises. This is Monday Tuesday, Wednesday Tuesday and Friday. Simple logic governs the criteria that I have given to you. That's it. Understood? So what are the two criteria here? The word every and the word unique. Remember this. Every and unique, if you remember, every is the one line explanation for what makes a relation a function. Understood? Now a few technical terms. Again, I'm coming from layman to not technical words. Because you will be getting to see them in your textbooks. There is something called domain. Have you heard of this word? Not that domain of your websites and all that. What's the domain, Aditya? Eh? No? Yes. Oh, sorry. Domain is x-axis. Ah, good. Again, let me just... Actually, I prefer a whiteboard, but today I have to show you some graphs. That's why this census is being used. Else, many of the classes will happen in the whiteboard only. What's the domain? There are some words that you'll come across. One is domain. Domain are basically those elements of set A which are getting mapped. Right now, I'm not talking in terms of functions. I'm talking in terms of relations. So, let's say I had a six over here. We are getting mapped. One, two, three and four. That's it, right? These elements would be called the domain. So, how is domain different from the pre-may set? Four and six. Only the ones which are getting mapped would be called domain. Understood? Understood. No, I'll explain it again. You're Neha, right? Yes. I'm sure you're not joking, right? I can't believe myself. Okay, remember? So, domain is basically the elements which are getting mapped. Getting mapped means they are participating in that relation. Right? So, those will only constitute the domain. However, all of them would constitute a pre-may set. Understood the difference? Pre-may set versus domain. Pre-may set will contain everything in setting. Whether it's the getting mapped or not getting mapped. But domain will contain only those ones which are getting mapped. Understood that? What is range? Range are those elements of set B which are getting mapped. Cut. Sausage. What do you think is the domain here? Sorry, range here. Is it too small? You want to go ahead? No. Roddak. What is the range here? Should I repeat the definition once again? Range are those which are getting mapped. Or which are participating in that relation. So, what are the ones which are participating in the relation? Four, five, six and seven. So, four, five. Five versus five. Six, seven. This is the range. Correct? Now, the distance you know between range and the image set. You will not make a difference. You will not make a mistake in that. Okay? So, who will stand up and explain to me? This is domain. This is pre-image set. This is range. This is image set. Who will stand up and explain to me? It's kind of, do you like to try? Sit and speak more. You want my mic? You will learn here? Yeah. A pre-image set is a set which is to be compared with the other one. Okay. So, here the set which contains your X is your pre-image. Okay. And what's that? What's that doing? It means the elements inside it are getting mapped. Absolutely. The elements inside A which are participating in that relation are getting mapped. Correct? Okay. Aditya, would you explain the difference between range and image set? Okay. All the elements with respect to Y. Okay. So, basically a set which contains... All the values. Which contains the set or which contains the elements of where Y is to be picked up. That is called the image set. But the Ys which are actually getting picked up or getting mapped, those will be called the... Understood the distance now? Okay. Now this is a definition which is more or less true even for a function. The only difference being... This was in terms of relation I was talking about. The only difference being in case of a function there is nothing called pre-image set. Everything is a domain. Why is that so? Because... Because why is that so? Because every participant there is no option given to them. You have to come and participate. So when they have to participate, doesn't the pre-image set itself becomes your domain? Understood. Understood? So 6 should not be there at all first of all. I am removing this. So 1, 2, 3, 4 is also the pre-image set and will also be the domain because of the criteria number 1 which I have written. Every element of the pre-image set must participate in the mapping. There is no option for them to be left out. If they do then you are not eligible to be called a function. Understood this? Clear? Shushant. Shushant right? Achievement. Okay. And in case of relation, in case of function this pre-image... This image set is given a different name. It's called co-domain. Just to rhyme with domain it is given a different name co-domain. What is the co-domain? Image set. So we say image set in case of relations but in case of functions the same thing is called a co-domain. Why it is called co-domain? Just to rhyme with the word domain. That's it. Correct? So... No. That's what I am going to come now. Co-domain are all the elements of the set B. But range which participate in the mapping. So here... Understood. What's your good name? Aditi. Aditi, Aditi. Oh my God. So many people have the same name in this class. Three Adityas, two Aditi's. Three Adityas. Yes, yes. The name co-domain is just point for functions. In relations we don't use the word co-domain. We use the word the image set. Is this a nomenclature as change? Functionality wise nothing has changed. Understood? Now, summing up this entire thing. Now you know the difference between pre-image set and domain. Do you know the difference between image set and co-domain? Actually there is no difference just in the name. But do you know the difference between co-domain and range now? Yeah. So five terms you have heard. Domain. Okay. I am again summarizing it. The word domain and co-domain are used in case of functions. Correct? In case of relations we use the word domain range. Okay. Something which I will again talk about in the chapter relations and functions.