 All right. So let's start the session on a relation and functions. It's a third session for you And I'm assuming that all of you were present in the previous two sessions. If you're not I would request you all to watch the videos already posted on the group So let's resume with today's session Dear students if I recall correctly last class I talked about one one function. So we are doing types of functions We were doing types of functions and in the type of functions. We started talking about one one function Okay, so types of functions and The first one under that was a one one function a one one function Okay, one one function. What was the definition of it? I don't worry. This is just a quick recap So if anybody is joining in late, okay, don't panic. I have not started with the session So this is just a quick recap of whatever we ended in the last class Last class. I remember correctly that I did equivalence relations with you we did a lot of questions on them and We started with types of functions and under that we were doing one one function. Yeah, so what's the one one function? Can anybody define me one one function? Just quickly unmute yourself Plain and simple language plain and simple language. What's a one one function a one one function is where your different different different images come from come from different pre-emages Come from different pre-emages just plain and simple. You don't have to break your head in understanding anything else So if an example to give an example of a one one relation, so let's say there is a function from set a to set b Okay, set a has let's say a bc Set a b has let's say one two three four. So this type of mapping Okay, I'm just giving you a simple example. This type of a mapping is called a one one function Okay, you also call it as an injective function Injective function some books will call it as injections Okay, it's a combination of injective and functions Okay, so you can see different images So one three and four are images over here. They come from different pre-emages Right. However, you would recall in the definition of a function. We had learned that two pre-emages can map to the same image Right, that is also a part of a function But in one one function, we do not entertain such cases where two pre-emages are pointing towards the same image Okay, so definition wise it is clear. No doubt about what's a one one function And if I recall correctly, I had also done how many one one functions are possible Number of one one functions So let's say if a set a contains N elements and if a set b contains m elements How many one one functions were possible? How many one one functions were possible? So we took various conditions first we took the case where m is less than n If m is less than n that means the number of elements in set b is lesser than the number of elements in set a Could we have any one one function possible? Yes or no. No, no There was zero one one function in those cases Where m is greater than equal to n the number is m p n p stands for permutation here I'm sure all of you know the formula of n pr just a quick recall from your last year chapter PNC chapter n pr formula is what n factorial by n minus r factorial Okay, so this is what we had done the last class So just quick five minutes. I took to recap it Any questions on this anybody? Okay So good morning to everybody who has joined in Hope you're doing well. Hope you are safe at home studying. Yeah Okay, so now we'll we'll go into more details of this one one function The first question that arises in your mind is how do we identify a one one function? If some function is given to me, how do I know that it's a one one function? So let's do an identification of one one function. Okay, so how do we identify a one one function? identify or prove Where there are function is one one function Identify an injective function Okay Now all of you, please Pay attention over here because this is a part which is going to be asked in your board exams also Okay, so just pay attention to this what I have to say if possible put your pen down I'll give you ample time to know no down things Now just try to understand these you know steps, which I'm going to tell you Now when you're going to identify a Function to be injective function. Basically, we can do it in three ways Right, there are three ways to identify Okay The first way is By drawing the graph of the function Okay, or you can say graphical method, but let me tell you this method is just a optional method Okay. Hi, Vinay. How are you? Vinay had accidentally on his video cam. No worries So graphical method is basically a method by which you can identify By plotting the graph or sketching the graph of the function But let me tell you this method is only a supplementary method Right in the board exams probably your examiner will not entertain Just the graphical method of proving a function is one one Another drawback with this method is many of us may not be very good in sketching graphs and Many at times the functions may be too complicated for us to sketch the graph Okay So nevertheless, I will tell you this method. What is this method so that when you are you know applying it to? Verify your result you can do it. So what does graphical method say it says that a function is one one a Function is one one if a horizontal line drawn a Horizontal line drawn What is the horizontal line mean horizontal line means a line parallel to the x-axis a Line parallel to the x-axis Okay, so if you draw a horizontal line It must not cut the function at more than one point Okay, a horizontal line Must not cut Must not cut Must not cut the function graph The functions graph at more than one point at more than one point Okay, now, let us understand why does this you know horizontal we call this as a horizontal line test We call this as a horizontal line test horizontal line test line test Now don't confuse it and don't confuse this with the vertical line test vertical line test is used to You know in and establish whether that given relation is a function or not Right if you recall the very definition of the function was Every image or every pre-image must have a unique image, right? This was the definition of a function, right? In order to establish whether a relation is a function or not. We used to have vertical line test. I Think you would have done this in class 11 Right everybody's clear about vertical line test I'm not talking about vertical line test here. I'm talking about horizontal line test Okay, so if you have been given any function, I'll just a sketch a you know Just rough figure. Okay, let's say my function is like this Okay, so this is my function graph Some arbitrary, you know function I have sketched if you draw a horizontal line If you draw a horizontal line, let's say this yellow line is a horizontal line You will see that this function will be cut only at one point by this horizontal lines only let's say at point a In that case this function will become a one-one function. So this becomes a one-one function, okay? If it is cut at more than one point Let me give you a typical example of that. Let's say I have a function like this. I have a function like this Okay, and if I draw a horizontal line like this, okay What is going to happen in this case? It is cutting at one two Three okay, so there are three points at which this function has been cut by this horizontal line in this case This function will be not a one-one function. It will be a not a one-one function Now the question arises why? Can anybody explain me why it happens? Why if it is cut at more than one point we don't call it as a one-one function any any answer See, it's very simple. It's very simple If you look at the second case, okay, what is happening? Let's say this is an output output is B or you can say output is Y Okay, this output is coming from three different inputs. Let's say I call it as a B and C Right, so what is happening is that okay? Let me name it as D Let me name it and let me name this output as D or O. Okay, O will be a good idea So this output O is coming from three separate inputs a b and c Okay That means if you're going to sketch an arrow diagram for such a scenario if you're going to sketch an arrow diagram For such a scenario what is going to happen is that? a, b, c will all point towards O correct So you can clearly see it's not a one-one scenario Okay, in one-one scenario every image should come from a different pre-image But this image O has come from three different pre-images right, so same Image has come from three different pre-images. That is not a one-one function by the very definition of a one-one function Getting my point, but what is happening in the first scenario? Let me name it as case one and case two But what is happening in the case one? Let's say this output. Let me name is as O dash is coming only from one input Which is a dash Okay, so let me call this function as g function So g a dash only is giving you O dash and no other pre-image is giving you O dash Getting my point. So this is this is accepted as a one-one function. This is not a one-one function Is that fine any question regarding horizontal line test again? I'm repeating it do not get confused with vertical line test Vertical line test is used to a certain or used to establish whether a given relation is a function or not Okay, but horizontal line test is used to a certain whether a given function is one one or not So there's a difference between these two Is this clear please type clear on your chat box so that I can proceed at least I should see 15-20 clears on my chat box Thank you so much my dear students for confirming this Now we'll move on to the second method now. We'll move on to the second method Which is the non-graphical method? Very good. Thank you Okay, second method that we are going to talk about is basically a non-graphical method Which we call as the uniqueness test which we call as the uniqueness test This is very important and Majorly, we are going to prove the one-one nature of a function All right only by these this test graph is supplementary. You can draw the graph to support your answer Okay, don't don't draw the graph and say okay see in the graph. That's a one more thing. I forgot to tell you When you are trying to prove the one-one nature of a function or trying to prove whether a function is one one Don't draw such a horizontal line, which will save it from cutting at any point Okay, you don't have to save the function for example, let's say There's a graph like this Okay, right is this function one one or not you don't have to draw a horizontal line here and Say sir see this horizontal line is not cutting the function at any place So it is a one-one function. No my dear you don't have to save the function You have to scrutinize the function Right, so you'll draw at such you'll draw at any such place where it is cutting it at more than one point Okay, if it does it will not be a one-one function Okay, so don't try to save the function Try to scrutinize the function. Okay, so don't do these kind of activities Okay So I was telling you that uniqueness test is the major test that we normally follow for proving a function is one one Graph is only there to support it Right, so what is uniqueness test now all of you please listen to this very very carefully See whenever a function is given to you, they will mention you the domain of the function So basically they will mention you the domain of the function And they will mention you the co-domain of the function, right? DF and CDF means domain and co-domain of the function correct Normally even if co-domain is not mentioned domain is definitely mentioned Let's say both are not mentioned Okay, then you can find your own exhaustive domain all of you know how to find domain last year You would have done that in the function. Okay in class 11 Function chapter only comprises of finding domain and range So many a times when they will give you a function I would say 99% of the time They will mention you the domain of the function at least if not the co-domain. Okay, mostly what do we do? We mentioned both of them to the students. Okay, so the function would be now given to you in this form This is the this is how the function is, you know written in the exam. Okay Now when you're checking whether this function is a one-one function or not, what do we do is we say that? Let's say there are two different Inputs x1 and x2 Coming from the domain of the function Okay, so what do we do? We we choose two inputs x1 and x2 from where? From the domain of the function Okay, such that such that F of x1 is equal to f of x2 f of x1 is equal to f of x2 Now try to understand what is happening You are assuming that there are two inputs x1 and x2 which give you the same output When you're writing this Basically, what are you assuming you're assuming that there exists? two inputs x1 and x2 Okay, which give you the same output which result into the same output Okay, you can say same image So these are two pre-images which result into the same image This is what you are assuming in the beginning of this proof Okay, and by using this I'm putting dot dot dot because it all depends upon what function is given to you and if by using this you end up showing that x1 equal to x2 is the only possibility Please note this word only possibility only possibility Well, let me explain this to you because this is Confusing at times if you don't listen properly you will get confused So request you to put down your pens. Okay, if possible do this to your ears No, I'm just joking and just listen to this carefully See what I'm trying to say is you start with two inputs x1 and x2 from the domain of the function Whatever is the domain given to in the question Such that the two inputs or what we call two pre-images Give me the same output or map to the same image and By assuming that you later on you know end up realizing that If such a thing is to happen it can only happen when the inputs are also the same What does it mean that means two different inputs cannot give me the same output Are you getting my point? Why only possibility only possibility is because when you are solving this kind of equation x1 equal to x2 will always come as a solution Trust me. We'll take some I know questions in today's session x1 equal to x2 will always come as a solution Right, so that doesn't mean every function in this world has become a one-one function Right apart from x1 equal to x2. There may be other relationship between x1 and x2 also Okay, if you are able to negate that or if you are able to prove that the other relation is not possible Only possibility is this one x1 equal to x2 Then only you have proved that the function is one one Let me give you an example one example is you know better than explaining it hundred times. So let's say Let us say Let us say I have a function From real numbers to real numbers You all of you understand the meaning of R with a double-danda double-danda means it's a it's a symbol that we use for real numbers Okay, and the function is defined like this The function is defined like this x2. Okay, normally this thing will be given to you by the examiner This will be given to you by the question set it Now, how will I proceed see the steps? I will say let x1 and x2 be two inputs taken from the domain of the function now This is the domain of the function Okay, such that such that F of x1 is equal to f of x2 So this is my assumption Right, so I have taken two such inputs, which will give me the same output Now if I say f of x1 equal to f of x2 indirectly I'm saying x1 square is equal to x2 square because my function itself is x square So if you put x1 there, it will give you x1 square if you put x2 there, it'll give you x2 square Now if you simplify this now now once one important step over here guys I'm just making a request to you. Please do not cancel out these powers from both the sides Please stop doing that if you have been doing it before Okay, I have seen student x1 square equal to x2 square square gone x1 equal to x2 Are you that doesn't work like that? It doesn't work like that. Let me tell you if you're doing that you are losing out on the roots Never cancel factors never cancel powers on both the sides Right. This is not a way to deal with this. Please Don't do this. Don't do this since your requests. Okay So how to deal with this The approach to deal with this is bring it to one side. So let it let x2 square come to one side Okay Since I don't have space. I'm just scrolling my screen to the right The factor is this like this. Oh