 We'll talk about few important concepts related to it. The first thing that I would like you to understand is an ordered pair and ordered pair. What is an ordered pair? If you say ordered pair a comma B, how is it different from a set containing a comma B? Okay, let us understand this first in order pair. The position of these terms cannot be changed. If you change their position, they will have a different meaning altogether. A classical example of an ordered pair is your coordinate system. So when you represent the position of a point, when you represent the position of the point on the XY Cartesian coordinate system, okay, let's say the point is 1 comma 3. This is an ordered pair. So if you put 3 comma 1, it would have a different meaning. The point will probably come over here. Whereas in a set, whereas in a set, let me write versus in a set. Even if you change the position of a and B, the set still remains the same. It is not affected by the position of the elements in the set, but in an ordered pair, you would realize that the position is sacrosanct. You cannot play with the positions. Okay, that's why the term ordered has been attached to it. Okay, so pair because there are two components to it. By the way, let me tell you this is called the first component or the first element. This is your first component. So when I'll be referring to it, I'll be referring it to as the first component of the pair and B is called the second component. We is called the second component. Okay, many books will call this as the first element and the second element both mean the same thing. Fine. So the formal definition of it will be and ordered pair is a pair of objects taken in a specific order and ordered pair. And what ordered pair is a pair of objects pair of objects taken in a specific order taken in a specific order. Okay, few things that we should know about ordered pair. Number one, if you say A comma B is equal to C comma D. That means if you're comparing to ordered pairs, it implies that you are saying A is equal to C and B is equal to D. That means the first component of the ordered pair on the left-hand side of the equation would be equated to the first component of the ordered pair on the right-hand side of the equation. Okay. And the second component of the left-hand side would be equated to the second component on the right-hand left-hand side. Fine. So as an example, if I give you that the ordered pair x comma y sorry, ordered pair x minus 1 comma y plus 3 is equal to 2 comma x plus 4 fine x and y. How will you do this? You will do you'll say it is very simple when you're equating these ordered pairs. It implies that x minus 1 should be equated to 2 and y plus 3 should be equated to x plus 4. Okay, let me call it as two different equations. So from the first one you end up getting x as from the first one you end up getting x as 3 and from the second one you get y as 4. So x is 3 y as 4. Okay, Tanishka is very easy. Fine. Let me give you one more on this find x and y if the ordered pair x square minus 3x comma y square plus 4 y is minus 2 comma 5. It's all for x and y it's easy, right? So when you're equating these ordered pairs, it is like saying x square minus 3x is negative 2. You can say it to be your first equation and y square plus 4 y is equal to 5. That's your second equation, right? So from the first equation, you end up getting x square minus 3x plus 2 is equal to 0 and this is factorizable. This is factorizable. So x x minus 1 negative 2x plus x minus 1 equal to 0. So x minus 2 times x minus 1 is equal to 0. So this will give you x value as 1 and 2, okay? In a similar way, if you equate this to 0, I'm sorry, if you get solved for this, so this minus this is 0. This is factorizable as x plus 5 into y minus 1 equal to 0. Okay, so y will be 1 or y would be negative 5. So these are your possible values of y. These are your possible values of y. Is that fine? Very good. Meanwhile, before I forget, I'll send you a theory come problem sheet. I hope you're working on it, right? So last class we completed with sets. So I think you will not be in a position to solve all those questions. So you don't have to do every question from there. Probably you can pick up the ones which you feel you might get stuck in or you can take odd number questions or even number questions, okay? And just try your hand at as many questions as you feel you are, you know, making as much as you feel that it is making you confident. Okay, you don't have to sit and do all the questions. Now in 11th and 12th, the game is slightly different. You have to be self-motivated. Okay, in the sense that when the class is over assignments have been provided you will have to practice as much as you feel is going to make you confident in that particular topic. In case you're stuck somewhere, please use the group. I think the other day Prisha asked one doubt. So you can always ask your doubt questions. You just take a pic and post it there. Either your peers will only help you or if I am available, I'm free. Okay. If there's no class for me, I'll be helping you out. Now, please let me tell you I am also a faculty at one of the schools of NPS. Okay, so I'm not available in the morning time. So my classes run up to 345. So then again, I have classes in the evening till sometimes 630 or 7 so post that I am free. So in that period only you can expect a response from me. Normally, I will I may not be able or early in the morning early, I will be able to respond to you. Okay. So when you posted, yes, who is this? So Prisha NPS classes started. NPS class 12th has started. Okay. Yeah. So which branch? All the branches have started. Oh, yeah. Yeah. So 20 today was the first class today was the first class, but I'm associated with half is cool. One of the schools in India. Okay. So normally my classes run from 8 to 345. So after that, you know, all of you know that sent them classes are there. So sent them classes will be there. So post 738, I'm normally free. So I'll respond to your messages. 11th has not yet started anywhere. Prisha. Okay. Okay, so many update on the NPS test. I'm sorry. Do you have any update on the NPS test? No, dear. I don't have any update, but let me ask few students so that they can because see everybody is now taking classes from their home. Okay. So what is happening is that they're not able to meet the admin stuff. But I think they're operating on half the strength. Some of the admin stuff would be there in the office. So try calling at the office. Well, you'll get to know some, you know, information there. No one picks up. Is there an official mail that you can write a mail? Yeah, no one replied. No one replies because I am associated with Nafal. Well, I have the contact of Nafal, this thing. And probably I can get the contact of Rajaji Nagar branch. But can you try for HSA branch? They say actually to be honest, it says that all the tests are on the same day. So like now they are, I think their priority is not changed. Their immediate attention is towards clearing of the board exams. They have not yet conducted the board exams properly. So I think their immediate attention is towards that. Okay. But let me just try my from my side. I'll try to see if I can find a connect through the admin of Rajaji Nagar branch to HSA branch. Fine. I'll try my best. Thank you. Okay. All right. So, yeah, so I was telling you so after seven thirty or eight o'clock only in the evening, I'll be in a position to answer your queries. Fine. Now the concept of ordered pairs leads to the concept of Cartesian product of sets Cartesian product of sets. Okay. So you already learned sets. So let's say there's a set a and there's a set B. Okay Cartesian product of two sets is written as a cross B. Okay. It is not a product of sets. It is actually a Cartesian product Cartesian product is something else. I'll define it for you. Then you'll understand what is what does it mean? Cartesian product of A and B basically is a set only. Okay. This set will contain all ordered pairs of the nature a comma B such that a will come from set a and B will come from set B. Let me give you an example to illustrate this. Let's say set a is made up of one comma two. Fine. Set B is made up of let's say three comma four. Okay. If I say write down the Cartesian product of set a and set B. So you will write all possible ordered pairs that you can make such that the first component comes from this set and the second component comes from set B. So how will you make it simple one comma three? Right? One comma four two comma three two comma four. Right? So it is a set but this set contains ordered pairs inside it not single single element as we had studied in our previous chapter. So what is a Cartesian product of two sets? It is actually a set and it is a set of ordered pairs and these ordered pairs are framed in such a way that the first component of the ordered pair comes from the set a and the second component comes from the set B. So this is how it is defined. Got the point? Is this explain this again? I didn't understand. Yeah, sure. See ordered pair Cartesian product of two sets A and B is basically a set of ordered pairs. Okay. So think as if think as if a and B are two families. Okay. So, you know, a and B are two families and you're taking the Cartesian product. So in Cartesian product, what will happen? You will have a set of all couples where the first person of the first member of the couple will come from the family a and the second member will come from family B. Are you getting my point? Okay. So this is an example. I'll give you one more example. Let's say your set a is made up of a comma B and said B is made up of 1,2,3. Okay. So when you say a cross B or Cartesian product of a and B you start making a set with ordered pairs such that the first member comes from set A and the second comes from B. So you can pair up a and one. You can pair up a and two. You can pair up a and three. You can pair up B and one. You can pair up B and two. You can pair up B and three. Right? So this is what we call as Cartesian product of a and B. Is that clear? Who asked that question? Pisha was that Veda not me. Me sir. I got it. Thank you. You got it. Yeah. So in the same example, if I ask you, tell me what will be B cross A? What will you say now B cross a means it will have an ordered pair such that first element will come from B and second element will come from a. So tell me what all ordered pairs will be there in the set. Somebody can unmute himself and say one comma A one comma B. Two comma A two comma B three comma A and three comma B. Now you would note that the number of elements in both these sets are same, but will you say the sets themselves are same? Will you say that set of A cross B and set of the same? The answer is no, no, because ordered pairs are not the same. You cannot say this and this are same. You cannot say five comma six and six comma five are the same coordinates. Okay. So few things that I would like to illustrate over here. Number one A cross B the number of elements are same as the number of elements in B cross a and that is actually given as the number of elements in a times the number of elements in B. So as you can see in this example, which I have cited A cross B has six elements because this has two elements. Oh, sorry and a this has two elements and this has got three elements. So two into three six elements would be there in a cross B. Okay. Second thing is commutative law doesn't hold good on this. That means a cross B is not B cross a that we have already stated is this fine? Okay. Now just generalizing it. You can also find I'm just generalizing it in the third point. Probably this will not be asked to you because it is slightly advanced level concept advanced level means I mean it is just a concept which is an extended version of what you are studying. So may not be asked by your school teacher. So if you generalize this you can have a Cartesian product of three sets also. So can you guess what would be this set be containing? Any guess it would be containing triplets. Right? So when there are two sets, you know involved in the Cartesian product, it will give you doublet. Doublet means pair, correct? It will give you triplets or you can say triples, whatever you want to call it. Okay. So these triplets would be such that the first of the triplet of the first component of the triplet will come from this set a second triplet of the triplet will come from this set B and the third of the triplet that is C will come from this set C. Okay. Just to give you an example of this a simple one simple example. I'll give you let's say a contains one B contains 2,3 and C contains let's say 5. Now if I ask you write down the Cartesian product of ABC. What will you write? So it will be said containing all triplets. Remember, so triplets can be 1,2,5 1,2,5 it can be 1,3,5 any other possibilities there? Any other possibilities there? No, I think we have exhausted it. Okay. We can't change the order. This element should come only from set A. This should only come from B. They should only come from C same for this. They should come from set A. They should come from set B. They should come from set C. So what do you use this for? Sorry. What do you use a triplet one for? See many a times what happens we will talk about 3D coordinate system in 3D coordinate system. You will have X, Y and Z coordinates for a point. Okay. Are you getting my point? So in this case your triplet would be used. There are quadruples numbers also. Doublets are also used. The pairs are also used in complex numbers. So the future will belong to those numbers which will have multiple dimensions. Single number is not sufficient for everything. For example, single number is only limited to expressing something on a one dimensional line. So we had to go for a double dimension number which actually helped us to scale up 2D figures. Right? Then we came to triplets where three dimension numbers were involved to signify any kind of a point present in space. Okay. Now I think quadruples have quadronians. What they call it quadronians people have also started working with quadronians. Okay. So maths is not a static field. Maths is something which is evolving every day. People are finding new theorems, new relations. So all those things are still under research. Okay. So the triplets concept is very rare. I don't think so your school will ever ask you this. Okay. And just to complete this the number of elements in A cross B cross C would be nothing but the product of the number of elements in A with number of elements in B with the number of elements in C. Is this fine? Any questions here? Okay. Now we'll speak about the graphical representation. Any questions? Please ask. No questions. Okay. Now we'll talk about the graphical representation of Cartesian product of two sets. Graphical representation of Cartesian product of two sets. Let me write off A cross B. Now there are two ways in which we represent the sets. One is by the use of a 2D graph. Okay. Where you plot the elements of set A on the x-axis and plot the elements of set B on the y-axis and show the components by the use by the by the you can say the lattice structure. Let me give you an example. Let's say I have one two three and I have a set B which is made up of two and four. Let's say. Okay. So what do we do is on the x-axis will show the components of the set A so one two three and on the y-axis will show the components of set B. That's it two and four. Okay. Now what we'll do is I'll make horizontal and vertical lines passing through these points as you would see that these horizontal and vertical lines will start meeting at these points. Let me show them with a different color at these points. Okay. And if you write down the coordinates of these points, it would basically represent your A cross B. For example, this point is one comma two. This point is two comma two. This point is three comma two. This point is one comma four. This point is two comma four. This point is three comma four. So if you write down the Cartesian product of A and B, you would realize you will end up writing the same pairs. So one and two, which is this point? Okay, one and four, which is this point? Sorry. Yeah, then two and two. Two and two is this point. Okay, then two and four. Two and four is this point. Then three and two. Three and two is this point and three and four, which is your this point. Okay. So this is a diagrammatic way or you can say graphical representation of A cross B. Okay. This this is basically called the lattice diagram. This is actually called the lattice diagram. Is this fine? This is very rare. We normally don't draw this. What we mostly draw is the next one, which I'm going to tell you, which is called the arrow diagram. Arrow diagram. So when we do relations in class 12th and all, we will be mostly dealing with arrow diagrams for representation of them. So how is arrow diagram made? It's very simple. We make two oval structures representing one as a set A and other representing as B. Inside this we write all the elements. Okay, and we show these pairings by the means of arrows. So one is paired with two. One is paired with four. Two is paired with two. Two is paired with four and three is paired with two and three is paired with three with four. Okay. So this is an arrow diagram now. You will understand the meaning of arrow diagram and all in a better way when you talk about relations. As of now we are talking about Cartesian product. Okay. So here everything is pointing towards everything else, but in relation you will not realize that or you will realize that only some of them are pointing towards some here. It is going from all elements to all elements, right? But it is not it will not be there or it may not be there in relations or functions. So but the starting point is this. So this is here where you understand what is a lattice diagram and what is the arrow diagram? Okay. It's a very simple concept. There's nothing to worry much about it. So if I say make a lattice diagram and arrow diagram for representing a cross B where a is given by let's say a comma B and B is represented by let's say alpha beta gamma. Okay. Now here you don't have to you know do anything. All you need to do is on your left hand side of the left hand side of the right hand side of the right hand side of the left hand side of the right hand side of the right hand you know do anything. All you need to do is on your x-axis. You just put your A and B somewhere A and B. Okay. And on your y-axis you put your set B elements which is alpha beta gamma. Okay, and just draw horizontal and vertical lines passing through them. So something like this wherever they meet right down the coordinates of that lattice points. So they will meet at six positions. You just have to write the coordinates of those lattice points. So this will be a comma alpha. This will be B comma alpha. This will be a comma beta. This will be B comma beta. This will be a comma gamma and this will be B comma gamma very simple nothing, you know very difficult about it if I draw an arrow diagram. All I need to do is make two overall structures. Put a would be write down the elements of a over here. Write down the elements of be over here start connecting them with arrows. So a will connect to these and similarly B will also connect to these nothing difficult about it any questions here. So next we are going to talk about some useful results. Some useful results. A few of them can come for your school exams for proof. First one. If you take the Cartesian product of a with a union of two sets. Okay, it follows this relation. Now basically it appears as if they are following distributive property. Can you prove them? Can you prove this? Yes, any idea? Okay, I'll just prove one for you and probably you can try another one on your own. See when you want to prove that a set is equal to another set. How do you prove it? How do you prove that a set P is equal to a set Q? How do you prove this anyone in the last chapter of sets? We did this right proving one set is equal to the other. Okay, I'll just help you recall this if you're proving set P is equal to set Q then you have to prove that P is a subset of Q and Q is a subset of P. Correct. If this is established that means this is automatically established clear? Okay, and how do you prove that as P is a subset of Q? How do you prove this you prove this? So in this case, we first take an arbitrary element which belongs to P and somehow if I prove that X will also belong to Q. Then only we can say P is a subset of Q. Okay, do you recall these two facts? Please recall these two facts. This is important because initially your school will ask you in the UTS and semester exams to prove one set is equal to the other. If you're not following this approach, then you will not be given the full marks. Okay, so now think as if this full set is your P set. See ultimately your Cartesian product sets only at the end of the day, they are sets. Okay, so this is a set. Let me call it as P for the timing and this is another set Q. So if I have to prove that P is equal to Q, I have to prove these two things to my examiner. What are the two things? I have to prove that P is a subset of Q and at the same time I have to prove Q is a subset of P. Is this fine? I say yes. No. Yes, sir. More of you are there and not only speaking, I can understand if it's a class of 40. Okay, else I feel I'm talking to my MacBook. Okay, so let me start with the fact that I'm proving that to show that a cross B union C is a subset of a cross B union a cross C. Okay, how will I do that? I'll first start with let's say there is a ordered pair a comma B which comes from the set a cross B union C. See, ultimately, what is my aim? My aim is I'll start with this and try to conclude somehow that a comma B also belongs to a cross B union a cross C. If I'm able to do this successfully, then this is proved. Okay, so now I'll be doing the same thing. Let me just erase it as of now. I've not established it, but this is my roadmap. This is the intention with which I am proceeding. Correct. Now, if I ordered pair is an member of Cartesian product of A and B union C. What does it mean? By the way, this is an arbitrary. Let me write that also because many people tend to get confused where a B is a comma B is an arbitrary the given set A cross B union C. Okay, so any arbitrary element or arbitrary ordered pair you can say to be more precise. Let's say it belongs to a cross B union C. Okay, so what does it mean? It means that a should come from a and B should come from B union C. Absolutely correct. Tanishka Tanishka. So a should come from a and B should come from B union C. That is from the very basic definition of Cartesian product. That means a should come from set A and B should belong to set B or B should belong to set C. Okay. If you open this up, it means A comes from set A and B comes from set B or A comes from set A and B comes from set C. Okay. Now individually, what does it mean? Individually, it means that A comma B comes from A cross B or A comma B comes from A cross C. Yes or no, I should not put unnecessary brackets. Is this fine? Anybody having any concerns, any doubts from jump from this step to this step and then from this step to this step? No. Clear? Yes sir. Okay. So we can say that A comma B is a member of A cross B union A cross C. Correct. And this is what I was aiming for. So I started with this step. All of you please pay attention here. I started with this step and I ended at this step. So what can I conclude here? I can conclude here that A cross B union C will be a subset of A cross B union A cross C. Okay. How did you get the last step? Sorry? How did you get the last step? Last step. See, OR means what? OR means union. Correct. So there's a pretty thing as if there is something written like this A belongs to P sorry X belongs to P or X belongs to Q. How would you infer this? X belongs to P union Q, right? In the same way, the role of X has been played by an ordered pair. Now, now the time has changed. Now we have upgraded ourselves not to single elements, but to double double elements. Okay. So this is an element which comes from this set or it comes from this set. That means that element comes from the union of these two sets. Yes, sorry. Now the proof is only half done here. We have to similarly show the examiner that A cross B union A cross C is a subset of A cross B union C. There you have to write the very same steps in reverse order. That is the trick I'm telling you. Okay. I think we had already done this. Remember last time it was right. So same thing. I will not do it. I'll just write similarly. Similarly, you may also show that A cross B union A cross C is a subset of A cross B union C. So if you're able to show these two things to the examiner, that means you have proven to the examiner that these two sets are equal. These two sets are equal. So I have given the proof here, but going forward, I may not give you the proof for all of them. But for a few of them, I may ask you to prove it. Any questions here? Anybody? No. Okay. So let me go to the next page. In a similar way, A cross B intersection C, you can write it as A cross B intersection A cross C. Again, you can think as if there is a distributive property followed of this product over intersection, just like the previous one. No need to prove it. Let me give you the third one. A cross B minus C is A cross B minus A cross C. Can you give me the half proof for this half proof means one sided because I know the other sided is just the same thing written in reverse fashion. All of you in your respective notebooks. Please give me the half proof for this half proof means please prove that A cross B minus C is a subset of A cross B minus A cross C. Try this out. That'll give you a practice of how to write it in the exam also done. Anybody? You're going to use two more minutes. Yeah. Yeah. Sure. Almost done. Okay. Okay. Take your time. No worries. Okay. Again, let's start with an arbitrary ordered pair. Which belongs to A cross B minus C. Okay. So let's say A comma B belongs to A cross B minus C. What does it mean? It means A comes from set A B comes from B minus C. Okay. The meaning of this is. A belongs to set A and B belongs to set B and B doesn't belong to set C. Yes, this is the meaning of A minus B. Right. Now this can be broken up as a belongs to set A and B belongs to B again and a belongs to set A and B doesn't belong to set C. Now the clear meaning over here is that a belongs to A cross B and sorry, sorry, my mistake A comma B A comma B A comma B belongs to A cross B and A comma B will not belong to A cross C because B is not coming from C. Correct. So what does this mean? Think as if now have a lighter picture in your mind. Then we'll able to connect to this faster. Let's say I'm saying X belongs to set P and I'm saying X does not belong to Q. What does it imply? X belongs to P minus Q. Right. Now this X role is being played by a comma B P role is being played by a cross B. Think like that. Okay. Similarly, X role is being played by a comma B ordered pair and Q role is being paid by a cross A cross C. Okay. So you can say that a comma B will belong to P minus Q. So P is your A cross B minus A cross C. Now this is something which you should not be writing. This is just for your understanding. I'm telling you. Okay. Don't write this in your school papers. Okay. From directly up to come from this step to this step. So what have you proven here? You have proven here that a comma B if it belongs to a set, it also belongs to another set clearly indicating that A cross B minus C is a subset of A cross B minus A cross C. Okay. Similarly, you can prove the other way. It is just the reverse order you have to write. Okay. Similarly, please prove that A cross B minus A cross C is a subset of A cross B minus C. Okay. So once these two are proven, you can directly say that this set will be equal to this set. Any questions here? So could you scroll down for a minute? Yeah, sure. Yeah, I'm done. And so. Okay. Good. Okay. So next thing that I would like you to prove here is if A and B are two empty non-empty sets, if A and B are two non-empty sets, that means they are not A and B are non-empty sets. That means they have some member in them. Okay. Then prove that A cross B is equal to B cross A implies and is implied by A equal to B. So you understand the meaning of this symbol? This meaning is the meaning of the symbol is if this is true, then this will be true. And if this is true, then this will be true. So vice versa. Are you getting my point? The symbol is called if and only if. If and only if. Many times we call it by a shortcut name IFF. Okay. So how do you prove this? That if A cross B is equal to B cross A, then A will be equal to B. Can we take an example? No, no, no, no. Proving cannot be done by example. You can just give me one half of the proof. You don't have to give me both the proofs. You can just prove here that if A cross B is equal to B cross A, then A is equal to B. That is sufficient. Just try it out. I'll help you or just give me a second. I'll unshare my screen for one second. Just give me a second. Let me help you with this. Let me help you with this. See, let's say I want to prove this. I want to prove that if A cross B is equal to B cross A, then A is equal to B. Okay. Now let me start with an element x coming from a set A. Okay. Let's say x is an arbitrary element of set A. Fine. Okay. So can I say and let's say B is an element of set B. So can I say x comma B? If it belongs to A cross B, x comma B will also belong to B cross A. Because it is given that A cross B is equal to B cross A. Let me remove these symbols here. Is this step understood? Okay. So what does it mean? It means x belongs to B. Correct. And of course, it means B belongs to A also. Right. But what is important is for you to understand that you started with x belonging to A. That's why I wrote it with an x so that you can make it out. And you concluded with the fact that x belongs to B. If such is the case, then we can say A is a subset of B. Am I right? Anybody having any doubt or any question on this? I didn't get the if part. See, I took an arbitrary element from set A and I took an arbitrary element from set B. Right. And let's say the statement is correct that x comma B will be a member of Cartesian product of A and B. Correct. Yeah. But it is also true that A cross B is equal to B cross A. So you can just replace this with B cross A. So by the definition x should come from B and this B should come from A. Okay. The moment you say x comes from B x comes from B and it also comes from A then A becomes a subset of B. Okay, got it. Correct. Now in a similar way, if you start with the fact that let's say y comes from B and let's say an element A comes from set A such that y comma A y comma A belongs to B cross A then from the given relation I can say that y comma A will also come from A comma A cross B because of this relation because these two are equal. Correct. So all I did was just replaced this with a cross B, which means y comes from set A and A comes from set B. So the focus element here is why so you started with y belonging to B and proved why is also belonging to a which clearly indicates that B is a subset of A. So from these two expressions from one into you can clearly conclude that A is equal to B. Is that fine? Any questions here? Now I will prove the other way round also. I will prove that if A is equal to B then A cross B will be equal to B cross A. So meanwhile, you please just note this down. Let me know once you're done. The proof is even simpler. So let's say to prove that if A is equal to B then A cross B is equal to B cross A. This proof is even more simpler. So if A is equal to B, okay. Can I take? Can I take cross product with B on both the sides? Or let me do one thing. Let me start with A equal to A. Okay, this is a universal truth. Now I'm taking cross product with B on both the sides. But while I'm taking cross product with B, I'm writing that B as A on this side because B and A are equal. Did you understand what did I do? First I began with let me start again. First I began with A equal to A. Okay, this is a universal truth. Two sets are equal to set two sets. One set is equal to itself. Now I took a cross product with B here, but I wrote this B as A in this side since A is equal to B it should not affect my result. Correct? Yes or no? Okay, now we know that B is equal to A. Let me call this as one. Now we know that B is equal to A. Let me take cross product with A on both the sides. Okay, so this will give me B cross A is A cross A. Right now since the right hand side of these two terms are equal, can I say their left hand side should also be equal? Right? So from one end to I can say A cross B is equal to B cross A. Done. Any questions here? Clear George Jacob? Yes sir. Try this one out. If A is a subset of B then show that A cross A will be a subset of A cross B intersection B cross A. If A is a subset of B show that A cross A will be a subset of A cross B intersection B cross A. Okay, let me handle this. See if you want to prove this is a subset of this, what will you do? You will start with an element of this set. Okay, so let me start with let A comma B be an arbitrary element of A cross A. Okay, so let A comma B be an arbitrary element arbitrary element means any random element of A cross A. So it means small a comes from set A and small b also comes from set A. Correct? Now have an eye on what do you want to prove? Have an eye on what do you want to prove? Okay, so what I can do is I know it will sound very very funny to you. I'll write the same thing like this a belongs to A B belongs to B. Okay, now I change this to B belongs to B. Why did I do that? Why did I do that? Because A is a subset of B. So any element of A will also belong to B. Yes or no? And I'm writing the same thing like this. I'm writing the same thing A belongs to A B belongs to A and now I am changing this element to A belongs to B. Okay, let me repeat this if you have not understood it. It would be better to write it, you know, in the way you will probably understand C. This thing is as good as repeating this twice. Does anybody object to this? This and the same thing, right? Nobody has any doubt in this, right? Now what I did here is I retained this term. I retained this term, but I changed this to be belonging to B. Why did I do this? The reason is A is a subset of B. So any element of A will also be an element of B. That is definitely going to be true. Correct. Now what I'm going to do here is since A is small A is an element of A capital A that is set A and A is a subset of B. I can change here also A with a B and others I'm retaining as such now this is I am doing this is a step which I am doing keeping the end result in my mind. So this is running in my mind that I have to reach this. Okay. So as you can see, I have actually created here A cross. Sorry, sorry A comma B is an element of A cross B and A comma B is an element of B cross A. Correct. Which means A comma B is an element of A cross B and means intersection B cross A right. So what have you done here? You started with a cross A and you ended up on this. That means you're saying this is a subset of this. So therefore A cross A is a subset of A cross B intersection B cross A. Sir, can you repeat how you got the intersection for it? Yeah, sure. I'll start from the beginning only. So I assumed an arbitrary ordered pair which comes from A cross A fine. So this means the first element comes from this set. The second element comes from this set. Correct. So this is what I wrote over here. That is from the very definition of A cross B. So now I repeated this two times. I hope from this to this. There is no doubt because I can always repeat it. Correct. Now what I did I had an eye on this expression actually. I wanted to get to this. So what I did instead of B belonging to a I made it B belonging to B. Now you must be wondering why did I do that? First of all, is it correct to do that? Yes, it is correct to do that because a is a subset of B. So any element which belongs to a will also belong to B. Correct. So it is correct to do this nothing wrong in that. And second reason why I did that is because I wanted to generate a cross B here. Okay, so this will lead to a comma B belonging to a cross B. Right? In a similar way in the second reputation which I did I changed this to a belonging to B. Because any element of a will also be an element of B because a is a subset of B. And this I kept it as such this was done because I wanted to generate B cross a got the point. So finally from these two bracketed expressions, I can conclude a comma B belongs to a cross B and a comma B also belongs to B cross a so a comma P should come from their respective intersections. That means a cross a is a subset of this. Clear Tanishka? Yes, sir. Now you don't have to do the reverse because you don't have to prove they're equal. You just have to prove this is a one-sided proof. So you just have to prove this is a subset. This will may not be a subset of this. Can you scroll down? Yeah, try proving this if a is a subset of B then prove that a cross C will be a subset of B cross C for any set C. Isn't this super easy? So if you want to prove this is a subset of this, we have to take an arbitrary element of this and show that that arbitrary element also belongs to B cross C. That is what we need to show, right? So have the roadmap clear in your mind. So let me start with an arbitrary element. Let's say a comma B or X comma Y whatever you want to call it. Okay, so let's say this is an arbitrary element which belongs to a cross C. So what does it mean? It means X belongs to A and Y belongs to C. Correct. No doubt about it. Now since A is a subset of B any element which will belong to A will also belong to B. So can I just change it to X belongs to B? Correct. Now anybody can guess this means X comma Y belongs to B cross C. So look what happened. You started with A cross C and you ended up on B cross C which clearly indicates that A cross C is a subset of B cross C done and it's proved. Is this clear? Okay. In a similar way, you may also try to prove this as a homework question. If A is a subset of B, and C is a subset of D, then prove that A cross C will be a subset of B cross D. Okay. Just try this for homework. It is super easy. You can do it in one shot. Now a very important property that I would like to take up over here. So can you go back and copy? Oh, so sorry. So sorry. We have not yet started with relations. We are stuck with Cartesian product only. I think just 10 more minutes. We'll start with relations. Done. Okay. For any four sets A, B, C, D A, B, C, D A cross B intersection C cross D is A intersection C cross B intersection D. Can you prove it? For any four sets A, B, C, D A cross B intersection C cross D will be equal to A intersection C cross B intersection D. Just give me half proof for it because other half, you know how to manage it. It's all using your and and or wherever possible. Okay. Not a very difficult thing. Okay. Let me help you with this. So first I would like to show that this is a subset of this. So let's show that A cross B intersection C cross D is a subset of a intersection C cross B intersection D. Okay. So I'll start with let there be an arbitrary element. You can say X comma Y which comes from a cross B intersection C cross D. So what does this mean? It means X comma Y is present in both A cross B and C cross D. Yes or no. So what does this mean again? X comes from a and Y comes from B X comes from C and Y comes from D which means X comes from a and X comes from C. Y comes from B and Y comes from D. So basically I just rearrange this. I just rearrange this. Okay, which clearly means X comes from a intersection C. Y comes from B intersection D. That means X comma Y will come from a intersection C cross B intersection D and this is what I wanted. So I started with this. I landed on this which means this is a subset of the last one. So I can say here clearly that A cross B intersection C cross D is a subset of a intersection C cross B intersection D. Any questions here? Similarly, you can prove the other way round. Okay. Similarly, prove that A intersection C cross B intersection D is a subset of A cross B intersection C cross D. Now, one very famous question comes on this property, but first you copy this down. Done copied? Yes, sir. Okay. If A and B are non-empty sets such that the number of elements common to them is n. Okay. Then prove that the number of elements, the number of elements common to A cross B and B cross A is n square. Can you prove this? Not by example. Can you give a generic proof? Okay. The proof for this is not difficult. Let's go back and look at the previous property. What are the previous property? Can you can anybody tell me A cross B intersection C cross D is equal to A intersection C cross B intersection D. Okay. So I'm just using this property. Okay. Now, what I'm going to do here is everybody please see here carefully. I'm going to replace C with B replace C with B and replace D with A. Okay. This is the universal. Can you guys hear me? Yeah. Yeah. Yeah. Sorry. Can you hear us? Yeah. Okay. So we can say the last like there was a power cut. So I was I was thrown out of the meeting. Okay. So yeah. So what I did was I use this property and in this property I repeat my C with and D with a A right. And I have to do everywhere. Here also I'll replace it with B and here also I'll replace it with a right. We can do it because it is universally true between any four sets A, B, C, D. So I'm going to rewrite this like this up till this. No issues. I have a question. Yes. How can you like replace C with B? I mean like what if they're not equal? No, I'm not making it equal. I'm just changing the name to be again. But then it's a little different. See, you can apply it to any force at A, B, C, D including C equal to B itself. Okay. It's like an identity. You can say that in identity you can play with your angles. Right. So when you learn basic trigonometric identity, sine square x is equal to cos square x. Okay. You can apply this identity even if your x is replaced with five x. It doesn't matter. Correct. Or if your x is replaced with x by two. Okay, you have taken it if B is equal to C, right? I have assumed my C set to be B itself. What if it isn't? No, no. I never said A, B, C, D should be different from each other. What if I do A cross A intersection A cross A? I can choose all of them to be same also. It is my call. Okay. Right. So it is true for any choice of A, B, C, D. So I chose my C and D to be B and A themselves. I can do that. Now you are thinking B and C should be different sets. Need not be. They may be the same. And A and D need not be different sets. They may be the same. Got it. Anishka. So basically you can have like the for four can be A, A, A, A. Why not? Yes. All of them can be B, B, B, B also. Okay. All right. Now, if I ask you how many elements are there in this set? Basically then you are counting how many elements are there in this set? Now remember number of elements in X cross Y is the number of elements in X into number of elements in Y. Correct? Now just keep a close eye because I'm now going to tell you something very important. Okay. Everybody is happy till this step. If not, please highlight here. Please let me know right now. You're going to give us a minute. I think of the last step. Last step. See when I was introducing Cartesian product to you, I told you number of elements in A cross B is what? Number of elements in A into number of elements in B. Correct? You remember this? Correct? Now just consider that this is your one set. Let's say A cross. This is another set B. So treat it like A and B respectively. Okay. So by the same property, I can write it like this. Correct? Yeah. Now this tells you something very important. This tells you by the way, my C was A. No. Why did I write C again? I'm so sorry. C was A again. Yeah. Now, this is something which we already know that the number of elements present in A intersection B. So come here number of elements present in A intersection B is N, right? So this number of elements here would be N and even B intersection A will be N so N into N. Now what is this left hand side denote left hand side denotes the number of elements common to A cross B and B cross A. That is what is the question asking me number of elements number of elements common to A cross B and B cross A. So they're actually asking us how many elements are there in A cross B intersection B cross A. That's what we obtained. So your answer is N square done proved. So do these proofs come for the exam? Few of them can come. Okay, but it all depends upon your school teacher also. Many school teacher will not care about these. They'll directly go to functions. So they'll not even ask single question on relations. Okay, it all depends upon how important the school teacher has, you know, attached to these concepts. See 11th, I'm telling you they'll do a lot of jumping and all the reason is 11th portions are not going to come for votes. But it is going to build up your basics for 12th. So they'll only talk about those concepts where your basics of 12th will be made up. For example, base relations functions sets. They will not even just see how I wish your how quickly the teacher will finish off sets because she knows it is not going to be asked in your board exams. Ultimately a school is I mean I'm saying this on record also ultimately that all the schools are worried about their board is a board results 11th is internal. 11th is an internal marking, right? So they'll only, you know, emphasize on few few topics which they feel is important for 12th like relation functions trigonometry. They'll they'll talk about limits and derivatives probability. So these topics will be again coming in 12th other topics. Of course they will do it, but they will not go into, you know, details of it like how I have gone, right? But you never know. You may have a teacher who is, you know, doing it in lot of details and they may ask you the proof for this. Initially when lot of syllabus is not covered and let's say immediate UT or semester exams is lined up. They may ask you questions in quite detail under this topic. So let's be prepared for the worst. Is this fine? Any questions? For any set, for any set A, B, C, okay. Prove that A cross B complement union C complements complement is A cross B intersection A cross C. There's one more, but I would just ask you to prove one of them. A cross B complement intersection C complement complement is A intersection B, sorry, A cross B union A cross C. Just prove one of them. The other one you can take down as a homework. Another important thing is that you have to be very, very meticulous with your performance with your assignments. Remember in the first class of which course I told you your success stands on a tripod stand. One is theory. Other is assignment solving and the third is testing. Okay. Theory and testing will be done by us assignment. Of course in an online mode, we cannot, I cannot check your assignment, but as a 17 year old or 16 year old, you have to be monitoring yourself. Tomorrow, let's say you go to a hostel where there's no monitoring done over you, you will be all by yourself. So you have to carry out all the learning part, your hygiene part, eating part on your own. Okay. Nine tenth was a different game when your syllabus was less, we could, you know, hand hold you a lot of things, but 11 12th hand holding is not possible. Okay. So logging of hours and all probably that is not possible because you should in fact waste no hour at all. So why there's a need to log hours that is fine for a ninth and a 10th and a seventh grader who is who doesn't know how to manage their time. But for a 17 year old who is about to achieve adolescence, logging of hours is a kiddish thing for you. You should not waste a single minute of your time. Whichever exam you are going to write, whether it's a CACPT commerce exam, whether it is a SAT exam, that should be your main focus. Of course, school exams will be also on priority list. So you need to chalk out a plan and work accordingly. Testing should also happen on a week by week basis. How do you do testing? I've already told you, for example, if you're prepared with a chapter X, you can always ask your sibling or your parents to give you, you know, a small section or let's say 20, 25 questions from that chapter, time yourself and then see what went wrong in that test. WWW analysis. So now start implementing this. Nobody is going to push you. And let me tell you after six or seven months into the system, you would realize if you're not following this, others have already overtaken you from my side. If you need any guidance, you can always give me a call, but hand holding is not possible. If you have a problem where you're stuck, I can always help you out there, but I cannot make and sit you make and do all the questions with you. It is nearly impossible because syllabus is very, very vast as I've shown you that 21 chapters to be covered. How many months to go? Just count today you are sitting on, let's say June, June, July, August, September, October, November, December, January, February, nine months, 21 chapters. That means almost three chapters or two chapters. You can say two and a half chapters per month has to be done. And you can see one set took me three classes. Okay. So we have to go. We still haven't started with relations. We still haven't started with functions. It is so vast done. Tanisha very good. Sorry. I'm calling you Tanisha every time is Tanishka, right? Tanishka. Okay. So it's quite simple. You can start with your left hand side. So a cross now, you know by D Morgan's law, you can write it as B compliments compliment intersection C compliment compliment that is going to be a cross B intersection C that this is something which we have already done. Right? I think it was the first of the second of our properties. So which is a cross B intersection A cross C done. Okay. Any questions here? One last question I would like to like you to solve and then I will move to relations. If a is a non-empty set, if a is a non-empty set such that a cross B is equal to a cross C. Then show that then show that B is equal to C then show that B is equal to C. Give me a generic proof. I don't want examples. Give me a generic proof. Okay, yes done. So let's say a cross B belongs to sorry a comma B belongs to a cross B. Okay. So what does this mean? It means a comes from set A and B comes from set B. Okay, but this element a cross A comma B will also come from a cross C because they are equal to each other. Which means a comes from set A and B comes from set C. Now just focus on B. Just focus on B. If B comes from set B and it also comes from set C. It means B is a subset of C. Right and you can repeat the procedure in reverse direction. So you can say let a comma B come from a cross C, which means a comes from set A and B comes from set C and it is known that a cross C is equal to a cross B, which means a comes from a and B comes from B. So if you focus on these two, it gives you an idea that C is a subset of B. So from one end to since both are subsets of each other, you can say B is equal to C done. Is it okay? Any questions here? No sir. Okay. Now don't be worried too much about this part, but please be worried about the part which I'm going to start with. Why I'm saying be worried because this is more important than that because the previous concept has led the way to understanding of relations. So this is what your school exam would be primarily revolving around of course board exam later on also. Now what is a relation? So up till now we did sets. We did Cartesian product offsets. Okay. Now what's the relation now relation as you all know the literal meaning is it is basically a property which connects to ordered pairs. Let's say I am related to you as your math teacher. Correct. So two objects are required. Okay. In fact, here two persons are required. Correct. So there is a relation between two objects. It could be any kind of relation. For example, your book and the cost of the book. It is the relation between them. Correct. Yes. I think he did not. Yes. I'm back. It's up. I don't know something is happening. It's going coming going coming five times. The power cut is happening. Okay. Anyways, so a relation is basically a property which connects two objects a and b. Okay. So we normally write it as a is related to be as a symbol for it and then we state the relationship between a and b. I'll give you a simple example for this. You understand it by that example. Let's say let's say m is a set of all men. Let's say in hsr. Okay. Set of all men living in hsr. Fine. And let's say w is a set of all women in hsr. Now when I say make a Cartesian product of m cross w. What do you do? In this you make a pair of men and women. All possible pairs of men and women where small m comes from set m. Yes, a screen got frozen for a minute. Okay. Can you hear me? So this is the Cartesian product of a and b. This is something which we have already done. Okay. But now I define something a relation which contains only those men and women. Of course, m should come from capital M and w should come from capital W. Such that m is the husband of m is the husband of w. Okay. So now what I'm doing I'm trying to relate a man and a woman in the relationship of them being husband and wife. Correct. So basically this is a relation which I have defined for you and let me tell you this is a rooster form. Remember rooster form. Okay. If you list out all possible husband wife couple in hsr you have basically written a sorry this is a set builder form. I'm so sorry. This is a set builder form. Okay. But if you list out all the couples you would have written down that particular relation in a rooster form. So what is relation? Relation is actually a set. Okay. It is a set of those ordered pairs which meet certain relationship. Right now. Does all man woman pair satisfy this relation? No, some some men may be these brothers of woman some man man maybe you can say children of some woman. Correct. So they can be so many relations that may exist between man and a woman. This man can be son of this woman. This man can be brother of this woman. This man can be father of this woman. Right. This man can be uncle of this woman. Right. So out of all the pairs see this is all possible pairs. So you are pairing up all possible pairs of man and woman. Okay. But only those pairs will be entertained in this relation which meet a certain characteristic defined by the question setter to you. Let me give you a mathematical example. This is a very you can say layman's way of understanding things. Let's say I give you a set a which has got one two three and I give you a set B which has got one four six. Okay. Now if I ask you what is a cross B what will you say? You will say simple it will contain one comma one one comma four one comma six two comma one two comma four two comma six three comma one three comma four three comma six. Correct. Now if I ask you this question or if I give you a scenario let's say there is a relation R which contains all ordered pair such that B is equal to square of a and this A comes from set A and B comes from set B. Okay. So I've given you a set builder form. Can you give me a rooster form for this? So which of these ordered pairs do you think satisfies this characteristic? Does one comma one satisfied is one the square of one. Yeah. Yeah. Does one comma four satisfied you'll say no. Does one comma six satisfies it? No. No. Does this satisfy it? Yes. Okay. Any other? No. So this is a relation which will contain only those ordered pairs out of A cross B which meet this characteristic or which meet this relationship. Right. So what am I trying to explain you over here relations at the end of the day is basically a subset of the Cartesian product of A cross B and this subset is decided on the basis of a characteristic given to you in the question. Is this understood plain and simple? Can you repeat that? Yeah. Sure. A relation is a subset of Cartesian product of A cross B where this subset is chosen on the basis of A and B satisfying or that's the elements A and B of that ordered pair satisfying a given relationship. Don't try to write this. Just understand this. Well, you'll be able. You'll get several examples for you to understand this. Don't worry about it. Okay. Let me give you another question in the same set A and B. Let's say I define another relation. Let's say I call this as R1. Let's say I define another relation R2. By the way, when you're writing R of relation, don't put double stroke double stroke is reserved for real numbers. Okay. Now you give me a rooster form. I'm giving you set builder form. Give me a relation which contains odd ordered pair such that A is less than B. And of course, let's say A comes from set A A comes from set A and B comes from set B. Can you give me the rooster form? Somebody can start telling me the elements ordered pairs. So is one less than one? No is one less than four. Yes. So one comma four can be there is one less than six. Yes. So one can one comma six can be there is to less than one. No is to less than four. Yes. So two comma four could be there. Two comma six could be there is three less than one. No is the less than four. Sorry. Yeah, is the less than four. Yes. So three comma four could be there and three comma six could be there. You will see for yourself that this is a subset of this. This is a super set. Right. So it has got almost nine elements, but this has only six elements. So this is a subset of a cross B and what are the basis of making the subset? The basis of making the subset is the relation which is defined by the user by the question setter. Is the idea clear? Yes, sir. So a cross B is playing the same role as the universal set used to play. Right. And are is basically playing the role of the subsets of that, but this are is not arbitrary, you know, a subset it is made on some kind of a relation and that's why this is called a relation. Okay, let me give you another example. In the same set A and B in the same set A and B. I define a relation like this which contains all ordered pairs says that a plus one is equal to B. Of course, a comes from set A and B comes from set B. Can you give me the roster form for this are three in case you want to see the elements. I'll just drag it down. Tell me in which of the case do you realize B is one more than a I think only in this it is happening. Right. In no other ordered pair B is one more than a right. So this will only contain three comma four done understood from these examples. What's the relation? So let me summarize this for you. Can you go back? Oh, so sorry. This relation could be anything. Let me tell you it is like, you know, it is in my hands how I want to, you know, define a relation. Done sir. Okay. So just to summarize whatever I have said in the last five minutes on relation. When you say there's a relation between set A to set B. Okay. Then representation for it is are is a subset from of A cross B. It is automatically understood that your first element of this relation will come from set A and second element of this relationship will come from B. Many books will also use this kind of representation. So please be aware of that. They will say R is a relation from set A to set B. Okay. Such that are is blah blah blah. I mean, I'm just writing related to be it could be anything. Okay. So this is something which is up to the examiner to fill. Okay. He will not say a comes from a and B comes from me. So that word which I used to write in the previous slide. That was only to explain you since you are new to this concept. Going forward, I would not write this part. I would not write this part. Okay. I would not write this part. Fine. So the moment I say there's a relation defined from set A to set B. It is automatically understood that this fellow will come from set A and this fellow will come from set B and only those pairs will come which are related enough given fashion. Are you getting my point? Okay. Let me give you more questions. Let's say A is a set of A is a set of all real numbers. Okay. There's a relation defined from a to itself. Now this is something again surprising. So you need not have two different sets to define a relation. You can define a relation from same set to itself. That means you can pick out two elements from the very same set which are related to each other by a given relationship. For example, let's say A is a set of all girls in my class. So girls in let's say maths class. Okay. I'm making a relation from A to A such that this contains two girls G1 and G2 such that G1 and G2 go to same school. Okay. So now Vedha, Prisha and Tanishka how many of you go to the same school? None of us. Oh, so this will be a null set. Okay. Let's say hypothetically let's say Tanisha and Prisha goes to NPSHSR and Vedha let's say goes to NPSCore Mangala. Then in this group only Tanisha will be paired up with Tanisha will be paired up with Prisha and Prisha will be paired up with Tanisha. So this I'm just assuming that let's say Tanisha goes to NPSHSR. Prisha also goes to NPSHSR and let's say Vedha goes to Core Mangala NPSCore Mangala. Okay. So this relation will only contain T comma P and P comma T. Now many people ask me sir do we have to write both? Okay. Yes, because now there's something which is coming on later for you something called symmetric relation and all. So as of now both of them will be put into the system. Okay. So this will be the roster form. This is the set builder form and this is the roster form. Okay. Is the idea clear? So there could be a relation formed on the same set to itself. So you need not have always two different sets to form a relationship between it. Fine. So now my question here is let's say R is a set of all ordered pair with satisfy this particular relation. Okay. Now I have given you a set builder form. I've given you a set builder form here. Can you give me the roster form for it? Everybody please write all ordered pairs. All order. Okay. Let me make your life simple. Let's make it integers. Let's say a is a set of all the integers. As they would be infinite such pairs. Okay. So I'm making your life easy. So let a be a set of all integers. There's a relation from z to z or you can say a to a and the relation is defined like this. This is the definition of the relation. Okay. Or you can say it is the set builder form of your relation. Okay. Now I'm asking you the roster form. I'm asking you the roster form. All the ideas that you have learned in sets will be applicable because see relation Cartesian product. They are all sets at the end of the day. Try to get this. But set having ordered pairs set having ordered pairs. So can you think of two integers such that their square is 25? You can say yes sir. Five comma zero zero comma five. Can I say zero comma minus five and minus five comma zero also can I say, three comma four can I say four comma three can I say Three comma minus four can I say minus three comma four. Can I say minus three comma minus four? Can I say minus Four comma minus three? Have a left out anything just let me know. Have a left out anything anybody anybody can point it out? So five zero zero five covered zero minus five minus five zero covered three four four three is covered three minus four minus four three is not covered minus four three is not okay so can I say only these twelve ordered pairs would be there sorry not twelve I think it's eleven anything that I'm missing out please highlight okay so this is the roster form for this correct now you can see that it is actually a subset of a cross a now a cross a will have infinite elements so I cannot sit and write all the ordered pairs having integers comma integer as their ordered pairs is the idea setting in is the idea setting in yes sir okay so let me give you another example let's say there's no end to this means I mean like if there's such a condition like where it can be like infinite answers then no for this they cannot be infinite answer this they can only be these many ordered pairs in the set can you give me any two integers other than the pairs which I mentioned which whose sum of squares will give you 25 no I didn't mean for this one I meant like for like any like general one they can go up to infinite right general one as what general one but what are you talking about so you limited this one to integers if you keep it like real numbers they if it is real numbers you'll get infinite answer but that's why I change it to integers because then you cannot know no teacher will ask you to write an infinite set don't worry okay okay let me frame another question so let's say a is 234 and B is let's say 123 okay fine now my question here is there's a relation from set A to set B by the way now I'll write it in some different way so that you get a taste of how you can also get a question in your exam so let's say this relation from set A to set B many teachers will also write it in words so there is a relation from it is read as relation from set A to set B from set A to set B and this relation is defined as all ordered pairs a comma B such that B divides a okay please write down the roster form for this so first question will be write down the roster form for this this at least I'm sure you are finding it easy right you may write your answer on the chat box to me so write all those ordered pairs where B divides a yeah write down those six Tanishka so it's too long or you can speak it out so Tanishka tell me what are the members there um 2 comma 1 2 comma 1 excellent 2 comma 2 2 comma 2 excellent 3 comma 1 3 comma 1 3 comma 3 3 comma 3 4 comma 1 4 comma 1 and 4 comma 2 4 comma 2 very good this is correct so she has mentioned see out of nine pairs possible remember a cross B will have nine pairs in it out of those nine pairs six of the pair satisfy the relationship mentioned to you in the question okay so those six pairs you have to list this down so as you can see the second member will always write the first one as per the requirement of that relation okay now in case of relation you can also draw your arrow diagram and lattice diagram so let's say if I ask you to make a lattice diagram for this how do I make it again in lattice diagram the elements of set A would be written over here 2 3 4 element of set B will be written over here 1 2 3 just make horizontal lines and make vertical lines as you can see this process is cumbersome that's why lattice form of representation is not that famous as compared to arrow diagram so only these will be shown so 2 comma 1 will be shown 2 comma 2 will be shown 3 comma 1 will be shown 3 comma 3 will be shown 4 comma 1 will be shown and 4 comma 2 will be shown so you can see that this this is not a part this is not included this is not included and this is not included okay now this is a very stupid way I would be very brunt over it this is a very stupid way to represent a relation a more acceptable way or a more commonly used way is where you mention it like this so write your a elements in this oval shape structure so what were the elements of a 2 3 4 right sorry 2 3 4 and elements of set B is 1 2 3 now only these six will be shown by the arrow diagram so as the nature pointed out 2 comma 1 so you'll make an arrow from 2 to 1 2 comma 2 you'll make an arrow from 2 to 2 3 comma 1 so you'll make an arrow from 3 comma 1 then 3 comma 3 and then 4 comma 2 4 comma 1 okay so your teacher may ask you to represent it by a lattice or a arrow diagram most probably she'll ask you arrow diagram only so this is more more commonly used this is commonly used is this fine so from this diagram you can say oh 2 and 1 are linked oh 2 and 2 are linked oh 3 and 1 are linked oh 3 and 3 are linked so this is just a diagrammatic way of writing the same thing okay so this is where you have used your listing or tabular form this is where you have used your diagram is this fine any questions here now there are two important things which we need to understand here one is the concept of domain and other is the concept of range I think we already spoke about it briefly in the bridge course but I'll be repeating it here once again so there's something called domain of a relation and there's something called range of a relation okay now how do you define domain of a relation domain of a relation would be defined as the first element all the first component of the ordered pairs which are participating in that given relation okay range is basically a set of the second component of those ordered pairs which are participating in that relation let me give you an example to illustrate this let's go back to the previous question previous question what was your set a 2 3 4 right said b was 1 2 3 correct and I asked you to write a relation where it contains all ordered pairs such that b divides a b divides a and when you listed it out you told me these are the ordered pairs 2 comma 1 3 comma 1 4 comma 1 2 comma 2 2 comma 2 4 comma 2 and 3 comma 3 right these are the six elements you said now all the first odd members or the first components of these ordered pairs okay if you put it in a set okay but remember in a set you cannot write any element more than once so the domain for this relation would be what it would be 2 3 4 will I write 2 again no because 2 is already there so domain will be 2 3 4 only which is actually in this case actually it is becoming your set a itself right but in general domain will be a subset of a please remember this in general domain will be a subset of a now what is range here range is the set of all the second components so you can see second components of 1 I think 1 2 and 3 all of them are participating okay but in general in general range would be a subset of b is the idea clear what is domain and range any questions here okay let's have a question to understand this in completion before we end the session today let's say a is a set containing all elements from one all natural numbers from 1 to 14 fine now there's a relation defined on set a now this is a new term that you're listening this only this means that this relation is a subset of a cross a okay so when your teacher writes there's a relation defined on a she actually means defined r is a subset of a cross a correct if she says it is defined from a to b it means it is a subset of a cross b okay now this is a definition or you can say set builder form for that relation it contains all ordered pairs a comma b such that 3 a minus b is 0 my question here is number one write the roster form for this relation number two write the write the domain of our and third write down the range of our everybody please do it we'll discuss in one minute a little more than one minute early are you done no sir take your time no worries done okay who will tell me the elements anybody any volunteer what ordered ordered pairs are there in our will tell me anybody Veda Prisha 1 comma 3 1 comma 3 very good 2 comma 6 very good 3 comma 9 very good and 4 comma 12 4 comma 12 very good I can't go 5 because 5 will give me 15 and 15 is not present yeah these will be the only 4 elements 1 comma 1 how 1 comma 1 is 3 equal to 1 oh sorry basically this is twice of the first should be equal to the second the price of the first component should be the second component okay so only these can be possible so now first part is done very good second part write down the domain of it so what will you do we'll only list out the first elements of this pair so 1 2 3 and 4 only will be the domain correct as you can clearly see this is a subset of a okay so here a and b both are a only right okay now what will be the range three six nine and one absolutely three six nine and 12 is this clear from this example any questions here now if I ask you make a roster form for this sorry make a arrow diagram for this so all you need to do is you'll have to write a and a in two oval structures actually it's one to 14 so I'll just try to write some of them so one two three four dot dot dot to 14 here also a light one two three dot dot dot six dot dot dot nine dot dot dot 12 okay so one will be connected to three two will be connected to six three will be connected to nine and four will be connected to 12 so these will be the only arrows you'll be making okay so even if you show this to somebody he'll be able to figure out what are the pairs can we have another question can you do another question okay so let's say a is a set of all natural numbers from 1 to 10 and there's a relation defined on set a okay how is the relation defined relation is defined like this it contains all ordered pairs a comma b such that 2 a plus b is equal to is equal to 10 write down number one the roster form for this number two state the domain of this relation number three state the range of this relation done okay so who will tell me the roster form one comma eight one comma eight very good two comma six two comma six very good three comma four three comma four very good four comma two four comma two very good any other no okay so what is the domain of this relation one two three four one two three four okay what is the range of this relation absolutely correct ten on ten understood you're finding this easy now yep much much easier as compared to those proofs that I was asking you Tanisha got panicked actually Adi is going to come in the school exams don't worry don't worry they won't ask you all those stuffs this is a very simple and scoring chapter actually to be very frank okay so let's have another question on this let's say let's say a is a set of all natural numbers from one to ten and there's a relation defined on set a and this is the definition of the relation it contains all ordered pair x comma y such that such that y is x plus five write down the rooster rooster form for this relation very simple question hardly should take 30 seconds for you none none so what's the answer one comma six seven three comma eight four comma nine and five comma five comma ten okay now I don't want to start a new thing because only just few minutes I left but let me tell you one slightly complicated aspect of relation all of these seems to be very simple very easy very manageable but all is not simple like that there may be some questions asked to you very would be asked to write relations where you are not relating to order two elements of an ordered pair but you are writing a relation having two ordered pairs themselves I'll give an example let's say there's just a you know scaled up version of the same concept let's say I give you a set n which is itself cross-product of natural numbers so natural number into natural number right and now I define a relation are on set a okay as this so this relation contains a ordered pair of two ordered pairs such that a plus d is equal to b plus c now this is something which is slightly complicated version of whatever we have done up till now our relations used to be on a single set right now this time the first time it has happened that I have given you a relation on a Cartesian product of sets what is the meaning of that see this will contain several ordered pairs right now this relation is a relation which connects to ordered pairs when how does it connect to ordered pairs by this relationship so any two ordered pairs which meet this characteristic would become a part of this relation are you getting my point see I'll tell you a simple analogy of it very simple analogy let's say today today let's say Prisha and Veda they are friends okay so there's a relation called friends and in that relation Prisha and Veda will be there okay let's say okay so this is a relation which is a friendship relation fine okay now tomorrow let's say after really 15-20 years Prisha will have a family so Prisha and family of Prisha okay will be friends of Veda and family of Veda okay so now what has happened the same relation has now gone into two ordered pairs are you getting my point are you getting my point so from single single aspect of relation now I have taken you to ordered pair ordered pairs just like I upgraded my set from a single single element set to a ordered pair set in the same way there could be relations written between two ordered pairs also let's say me and my friend let's say Tushar sir is my very good friend so me and Tushar sir are friends and we were friends since IIT days right now of course both of us got married so me and my wife are good friends of Tushar sir and Tushar sir's wife okay so the same friendship relation has now gone between two ordered pairs thing like that does it make sense to you or is it like still beyond your imagination the examples really helped yeah right so yeah so these examples will help you realize that there could be complicated versions of relations where you could have relations between not to single single elements but to ordered pair ordered pair also and later on this can be scaled up to you know triplets triplets also so let I can say me my wife and my daughter we are friends of Tushar sir Tushar sir's wife Tushar sir's son we can say like that it can be between triplets also okay now I would not like to go into any questions in the last two minutes because I don't want to want you to leave the session with some doubt in your mind we will start this fresh this concept will take few examples on them also so this is the toughest your teacher can make the concept she cannot go beyond this okay so I think next class is what we would need to complete a relation and start with functions relation there's not much in it so we'll start with function straight away function is slightly tricky it'll contain some concepts which we need to you know understand to great depth like domain range and all so we'll talk about it when we meet the next class so summarizing what we did today we talked about ordered pairs and ordered pairs as you know it is different from sets because the first component in the second component there is an order assigned to it we cannot mess up with the order secondly when two ordered pairs are compared the first component are compared and the second component are compared individually then we scaled up the concept to Cartesian product of two sets okay and in Cartesian product we saw a lot of properties and we proved some of the properties not all of them are important but few of them can come in your exam then we talked about what is a relation actually so relation is a subset of Cartesian product of two sets where an element of set a is related to some element of set b by a given relationship and we also talked about the concept of domain and range of that relation and towards the concluding part we also told that we also discussed that there could be a scaled up version of relations so there could be relationships not between not only between single single elements a comma b but it could be between two ordered pairs themselves okay by the way just a concluding word for me this is not how the teacher will write it the teacher will always write it like this the same thing would be written in this way okay so they're just different notations for the same thing but you will find your books are using this notation not this not this notation this is just for you to understand that it is an ordered pair of ordered pairs okay so thank you so much can you all suggest me thank you sir yeah can you all suggest me a time in the evening latest when you can have a class or Saturday Sunday Saturday Sunday I'm free okay fine everybody's free take some engagement in the morning hours but let's say 11 o'clock ish everybody free yes sir see I'm not saying anything right now I'm just taking your views probably I'll communicate over the group also fine next Wednesday can we keep it like from three to six or something I see as I told you three three 45 I'm supposed to be in in the session of the school so how can I be available so my morning my morning eight two three 45 I'm in the