CLASS 12 MATHS | RELATION FUNCTION | ALL IMPORTANT QUESTIONS | ONE SHOT

Ajouté : 2026-05-30

[00:00:20]Yes children, Namaskar, good evening, Sat Shrilal, Aadab, Jai Shri Ram, Ram Ram, Radheshyam, children, how are you, how are you and once again, welcome to my channel Vijay Sharma Math Lecture. So, son, today's important lecture is Relation and Function. There is a test tomorrow, so do it well. Then don't say that Vijay Sir did not tell you. Okay, I will try to revise all the questions for you. Okay, you did it many times before, we also revised it in the class. Okay, we did it many times, so we are okay. This lecture will be held in two shifts. First, we will study at 8:30.

[00:00:59]Okay? After that it will start again at 10:00 pm.

[00:01:02]Ok? See again. First it will be till 8:30, then it will be till 10:00 in the night. Ok? The second shift link will remain the same. So do it then son, start. Let's start the game. Understand it well and then do it. Then we do it.

[00:01:14]ok sir. Look son, if you are found absent then I will remove you.

[00:01:20]I told you first. I will remove it.

[00:01:22]Remove. Absolutely. Son, take attendance only when I say, not now, take attendance when I say, okay, so son, next, let us first do this question, xy, see, two- three questions very variant, okay, x - y is divisible by 3, right, so this is a relation on the set of integers, Garima, keep in mind that on what basis does this divisibility happen, on the set of integers, it comes in two ways, one is your x - y, simple one is mode, we will get both done properly. Do it carefully.

[00:02:27]First, prove that the relation R is but R is an equivalence relation on Z. Ok?

[00:02:35]clear? So do you know what? First of all, understand what is an identical relationship? First we have to prove it reflexively. What to prove?

[00:02:44]Reflexive. Now what will be reflexive son? Tell. Well done.

[00:02:50]For reflexive it is a simple thing, for reflexivity for all x belongs to z means take any x integer of this z and x belongs to r, first thing is that what should be the relation of each element with itself, that is reflexivity, what is symmetry, if x y belongs to r then it implies that y x belongs to r, then understand if xy belongs to r means x has a relation with y, then son in that case y also has a relation with x, then that is symmetry, what is symmetry, okay, what is transitivity, if x y belongs to r, right, if x has a relation with y and y has a relation with z, then son, then what should be that, what should I say, okay, okay, okay, yes, thank you, thank you so much, son, okay, tell me your name, you are our student, I mean, four years old, okay. You are fine, okay, so Doctor Sanjeev ji is my very fast friend, okay, the one who was your teacher at that time, he is my very fast friend, he is from Barnala, right, Doctor Sandeep ji, very very intelligent, great son, great, okay, ji, x belongs to r and y z belongs to r, employee x z belongs to r, right, okay, so the first thing is that if it is okay, you are fine, right, okay, Jatin son, where are you from? Kuldeep Singh okay Dhillu Kuldeep Singh okay first of all if x belongs to r then for all x belongs to which z is defined then son that is reflexive right if xy belongs to r implies y x belongs to r son what is that symmetry if xy belongs to r and what do you say yz belongs to r then that implies what do you say xz belongs to r so that is transitive what is transitive okay okay so okay now see carefully how do we prove it see this is reflexive first see what is reflexive if x belongs to r for all x belongs to this we do you know what we do here a if set if any relation r is A relation on A. Relation on A Okay.

[00:05:53]So son, if XX belongs to R then for all X belongs to A came here. A will come here.

[00:06:00]ok sir. So what will that relation be? Ok? Tell me sir. Reflexive. What will happen?

[00:06:04]Reflexive. Any doubt?

[00:06:08]ok sir.

[00:06:11]Yes. If x y belongs to r, if yz belongs to r, then what will be that relation, it will be symmetric, what will be symmetric, now look carefully, okay, so any doubt, butler, okay, now transitive, if x y belongs to r, then look if x y belongs to r, if xy belongs to r and y z belongs to r, if xz belongs to r, then that is transitive, three things have become, okay, then look carefully, first reflexive, if x x belongs to r for all x belongs to a, on which that relation is defined, then son, what has happened, that has become reflexive, if x y belongs to r, if y x belongs to r, sorry, y x will come here, y x belongs to, so son, what will that be? Symmetric. What will happen? Symmetric.

[00:07:14]Ok? The meaning is simple. If son, see xy belongs to r and implies y x belongs to r, then it will be symmetric.

[00:07:31]Ok? clear? So next if xy belongs to r and yz belongs to r then imply xz belongs to r ok clear so that will be your transitive what will be transitive kids have doubt ok next next now first reflexion ok ji sin x - x then see sin x - x = 0 which is divisible by 3 for all x belongs to z there for the relation r is reflexion. You know what x- x is, right? It is equal to 0, this is your division by three. What happens?

[00:08:56]What happens with three? Divide. What this means is that x belongs to r for all x belongs to z first for all x belongs to z first is done second son we have taken symmetric relation let xy belongs to r let xy belongs to r okay let it be clear so let this imply x - y is divisible by 3 see it is a very simple thing if your x belongs to r what is the relation r?

[00:09:28]What is the relation, son? Do you know what that relation is? xy is true that x - y is divisible by what do you say? Three. ok sir? So this is three. If any number is divisible by three, then son, what will definitely be that number? That is equal to 3 lambda will come. Where lambda belongs to z. You know 3 * something.

[00:09:54]Any doubt? Ok? What has happened now?

[00:09:57]How much did this employee earn y - x? It is equal to tell me how much you have got son? Yes, how much did y - x come? 3 * - Lambda and Lambda belongs to z Now what happened son? y-x also what did you get from three? Multiple arrived. Have you arrived? Multiple. This implies y - x is divisible by three. This employee y x belongs to r. There for the relation r is symmetric relation. Why?

[00:10:24]Because you have already completed writing xy belongs to r.

[00:10:28]xy belongs to r what did it imply say ji y x belongs to r if this is the case beta ji what is a relation symmetric yes ji Garima Ritika Prachi Payal Navi Butler Kartik Rohan and where is Rohan Noorin now see what was the first one sin x x now see attention sin x x = 0 w is dible by for all y 3 by what na y 3 for all x belongs to z there for the relation r is reflexive relation r is reflexive relation ok clear any doubt ok ji next look carefully do you know what you took first? xy belongs to r what took? xy belongs to r this employee x - y is what has come? Divisible by 3 This implies x - y. How much did it come to? What is equal to 3 times y - x? 3 * - Lambda no problem. Did y-x also come?

[00:11:48]Divisible by three came. So doubt there for y x belongs to r. What is the relation there, tell me? Symmetric relation. Now look what is number three transitive? Look carefully, let xy belongs to r, we will start from here and yz belongs to r. Okay, exactly, what does this mean? Your x - y is also divisible by three.

[00:12:14]And what does y - z also mean, that is also divisible by three. So there for x - y, how much does it mean? It is 3λ1 and how much does y - z mean?

[00:12:23]3λ2. Look, it is a very simple thing that xy belongs to r only when your x - y will be divisible by three. And yz belongs to r only when y - z will also be divisible by three. That is clear. Okay, so there for x - y + y - z is equal to what is 3λ1 + λ2 where λ1 + λ2 belongs to z. This implies y is cut off from y. This implies x - z = 3λ where lambda =λ1 + lambda2 belongs to z This implies x - z is divisible by three This implies xz belongs to r There for the relation r is transitive relation since r satisfies reflexivity symmetry and transitivity It is an equal relation on z So simple no problem now tell me if anyone has any doubt Lovedeep. Lovedeep Lovedeep Yes Sharma okay? Now look at these, these satisfy reflexivity, symmetry, and transitivity. It is an equals relation on Z. Look, what will be the relation? Good Sharma, it is absolutely reflexive, symmetric and transitive as well.

[00:14:21]What will that relation be automatically? It is an equal relation.

[00:14:27]Son, you do not say mda, assume m.

[00:14:31]Now tell me, tell me, what will any number divisible by three be like three will be written as 3 * 1 which is six will be written as 3 * 2 which is nine will be written as 3 * 3 which is 12 will be written as 3 * 4 which is 15 will be written as 3 * 5 which is 18 will be written as 3 * 6. This means that any number divisible by three will be written as 3 * something. Three is fixed, along with it something will come and that something should be an integer. We consider that something as lambda.

[00:15:06]If you do not like lambda, son, you can assume m. What is the problem? Okay, you can assume m. You can assume n. But it will be an integer only. 21 is 3 * 7. 24 is 3 * It is 8, which means whatever comes with three that should be an integer, we consider that as lambda.

[00:15:25]If you do not like lambda, do not know how to say it or write it, son, you can also consider it as m or n, it depends on you. Is it okay son?

[00:15:57]Mukesh Arora ji, where are you from son?

[00:15:59]Mukesh Arora ji, where are you from?

[00:16:18]Now look, now this is the question, I know what the difference is, it has mode, it's just a matter of how to play with it, you see, okay, first since x - x = 0 which is divisible by 6 for all x belongs z for the relation R is an R is flex relation. That's clear because look what it is? The relation of X with Y will come when what will be your X - Y mode? Is divisible by what be? That's six. Ok?

[00:16:55]What does it mean?

[00:17:02]Then look, now look.

[00:17:06]Ludhiana what is your son's name what is Mukesh ji your ID what is your name there for address what x y j belongs to r then when what comes your x - y mode is divisible by 6 will be ok. Ok. What is your own name, son? It is of ID. Ok. Ok. So what is x - x? What happens to the numbers 0 and zero? There is a divide.

[00:17:47]What does this mean? That your x x belongs to r has arrived. For all x belongs to z. What is there for this relation, tell me? Reflexive relation. What is?

[00:17:59]Reflexive relation.

[00:18:01]Ok? Number two.

[00:18:04]So the next symmetric relation is clear.

[00:18:07]Let xy belong to r. This implies that x - y is divisible by six. ok sir?

[00:18:16]Look at this, there was a mode earlier, then we saw the mode, one thing is certain. If there is any x - y mode and it is divisible by 6 then son, x - y will also be divisible by 6. sure will.

[00:18:28]No doubt. There is no doubt in this.

[00:18:30]Okay Chetna son, okay, okay Chetna, okay, this implies x - y is okay, this implies x - y how much has it come to, it is equal to 6λ, what has it come to, 6λ lambda belongs to z, this implies y - x how much has it come to, 6λ lambda belongs to z, same process. This implies y - x is divisible by 6. If this is divisible by six, what is y - x? It will be divided by six.

[00:19:00]Son, understand one thing carefully. There is no problem in this. The question is exactly the same as the previous one. But you should first look at the mode, if your x - y mode is divisible by 6 then definitely son x - y will also be divisible by 6. And the second thing is that if your x - y is divisible by six then x - y mode will also be divisible by six. In this you just take this much that let xy belongs to r this implies x - y mode is divisible by 6 this implies x - y you first remove the mode x - y is also divisible by 6 this implies x - y how much is 6λ this implies y - x how much is it equal to 6 - lambda so this implies y - x is also divisible by 6 so this implies y - x mode is also divisible by 6 okay it is clear so what is y x belongs to r there for the relation r is symmetry see this is very simple that x y belongs to r so what is it, you can say that y x belongs to r so this implies the relation r is what is it, you can say symmetric relation okay next transformation now look at this carefully there is nothing in it first you Read it and you will know that let xy belongs to r and yz belongs to r. This implies that x - y mode is double by 6 and y - z mode is also double by six. No problem, then from mode we come directly to simple. Yes, here too we have come to simple. So this is exactly the same step.

[00:21:02]x - y = 6λ1 and y - z = 6λ2, right? 6λ2 has come here λ2 so what has come? Then we added it. So what is this x - z? It is = 6λ right? So what has come to There For? x y z is the same step. After this step, the entire previous question started. Totally previous started.

[00:21:26]Previous Totally. ok sir? So what has come? is divisible by six. This implies x - z mode is by six. This x z belongs to r so what has come? What did there for xy belongs to r and yz belongs to r do? Employee implies xz belongs to r the relation r is what say it is ok equal session is simple someone look carefully first you took let xy belongs to r and yz belongs to r this employ first what will come x - y mode is what will come son ji by 6 and y - z mode is by 6 this employ then what do you have to do simple x - y is divisible by 6 and what came say y - z without mode is divisible by 6 so what came x - y = 6λ1 and y - z son those who are from other district please tell your name those who are watching from different district please tell your name λ1 and write your name your district son good evening Jiya Thakur how are you son x - y one of our meritorious student no in Chitkara why Jiya x - z = 6λ1 + λ2 this Imply x - z = 6λ where mbda =λ1 +λ2 belongs to z Speak up son ok Jasmeet Kaur from Ludhiana Patiala Jasmeet today I have received your message address Jasmeet I have received your message for the first time today Jasmeet Kaur how are you son? So, has this employee come?

[00:24:47]x - z is divisible by 6 then you will write that x - z mod is by 6 this implies xz belongs to r there for xz belongs to r so there for x belongs to r and yz belongs to r implies xz belongs to r any doubt son ok now these questions these questions are very variant this last year paper me aaya tha this baar na ye aaya tha aaya tha now it came in the paper in March 2025.

[00:25:38]Now, do you know what you remember after seeing this? Tell me son, son, children, say well done.

[00:25:53]What do you remember after seeing this question?

[00:26:12]Oh, there is no 1032, whoever it is, it is wrong, wrong, 10:43 absolutely, 10:43 absolutely, 10:43 absolutely, it is absolutely correct, 10:43 well done, well done, well done, well done, Garima very good, Sunaina, well done, well done, now look at the solution, the first scene, now look carefully, this is nothing, son, keep one thing in mind that this is nothing, neither is it reflexive, nor is it symmetric, nor is it transitive, and to disprove, look, then listen, to disprove, we have to take some example for which it is false, then to disprove Sunaina, we have to take some example for which it is false, and look at these examples carefully, now the children who will ask what is 10:43, okay, absolutely Jasmeet, you will understand, don't worry, you will understand right away, don't worry, okay, and after this, today we will also do MCQs, okay, of these, okay, so son, don't Okay, now this is not reflexive, there is a different example for that.

[00:27:31]This is not symmetrically symmetric, for that we have a different example. And this is not transitive either, for that we have a separate example.

[00:27:41]We have to disprove this by giving examples. It is neither reflexive nor symmetric nor transitive. Jasmeet son, now you will understand what this 10 43 is. It is a very useful thing. ok sir.

[00:28:12]Okay, first look carefully since 1/2 is not less < = 1/2², absolutely Mamta, okay, do you know what this 1 look is?

[00:28:28]Actually, reflexive happens when the relation of x comes with x for all of them. But if one looks at what is the relation? The relation of x with y will be when x is less than y².

[00:28:43]Now the relation of x with x will be when your x is less than x². But you see it is 1/2. If I take this x then 1/2 is never less than 1/2². Is more. Look, 1/2 is your 5. Yes, but it is 1/4.

[00:29:04]This is 25. This is wrong, isn't it? This is not less. What this means is that 1/2 1/2 does not belong to r.

[00:29:13]not doing.

[00:29:16]Ok. Clear, okay sir.

[00:29:19]Ok. So there for the relation R is not reflexive Jashan Choudhary you will be removed remove Monica son great great first of all you got 100 by 100 marks very great son what is your name and where are you from son what is your name and where are you from tell me okay and in which sessions are you elder than us?

[00:30:05]ok sir. There is the relation R is not reflexive. R is not reflexive. This is a very simple thing. See, is it clear? Because look at 1/2, 1/2 is not less than 1/2², it is a simple thing because 1/2 is 5, so 1/2² is 255. Is it clear son? Absolutely, you can take anything, any number which is between 0 and one, your number which is between zero and one, take anything because there are such numbers between 0 and one that if you square them, the more power you raise to, the smaller the number will become, you take one, that means you took 1/10, what is the problem, 1/10 is not less than 1/10², it is a simple thing, what does one mean, it is 1/10, we took 5. It's the same thing. Ok? That's absolutely correct.

[00:31:23]Navjot son, absolutely, your madam, madam Pur Hira, I know that madam is very very intelligent. Okay, she is also very nice and very intelligent madam. Okay your madam.

[00:31:48]Next Next Look Next Now look since one is less < 4² but 4 is not less than one square, look carefully the one which is yours is less than 4 squares. Is it clear?

[00:32:10]But what do you say? 4 is not less than 1² He is not. So what did you say there? So 1 4 is your belongs to R but 4 1 does belong to R and the relation R is not symmetric.

[00:32:31]Ok. Ok. clear? So what has come? The relation R is not symmetric no doubt ok Monica from Shri Mukh ok ok Monica ok ok son you are from Musa sahab yes 23 24 it is guessed I found out ok ok ok you are Batra sir's student na, Shivinder Batra sir's school na, God bless you son this Monica son God bless you ok so thank you so much that you have still subscribed to our channel.

[00:33:06]Thank you so much son. And God bless you son.

[00:33:31]Okay okay okay okay Monica okay okay okay okay yes okay okay okay that batchmate of yours is enough okay okay son now see the relation R is not symmetric.

[00:34:09]Next Next now see, it will be useful for you 10 43 ok 10 43 10 43 see Jasmeet son why did we take this, first two examples are clear, okay sir, so now see 10, it is less < = 4², first thing is clear and 4 is less < 3², this means that 10 has a relation with four. Is it clear? And four is related to three. That's absolutely correct. But 10 is not less than 3² 10 is not less than 3² Okay. Ok there, what do you say? tan 3 does not belong to r son, son, first tell me, look carefully that tan is a relation of tan 4. That's absolutely correct. Why is it? Because 10s < = 4² is correct. And 4 is a relation of 3.

[00:35:12]Why do you say it? Why? Because 4 of four is less than 3². But but what do you say? 10 3 is not a relation.

[00:35:24]Why not? What is the reason? Because 10 is not less than 3². That's why we remembered 10 43.

[00:35:33]Why was it kept? That first 10 is related to four and four is related to three but 10 is not related to three.

[00:35:42]Ok? This example is a little difficult to make on paper. That's why we already said what do you say. ok sir.

[00:35:48]Any doubt?

[00:35:51]ok sir. Yes. Ok. 10 is related to four.

[00:36:00]Ok. And four is related to three.

[00:36:02]But 10 is not related to three. Any doubt? Remember this example well.

[00:36:08]So son, now when you take such a relation xy, true that x < y² xy belongs to r, then that relation is nothing. Neither is it reflexive, son, it is symmetric, nor is it transitive.

[00:36:26]For reflexive, we took the example 1/2 1/2. For symmetry, we took the example that one is less than 4², but four is not less than 1², meaning one has a relation with four, but four does not have a relation with one. Third, what we took for transitive is that 10 is related to four and four is related to three.

[00:36:46]But 10 is not related to three.

[00:36:48]For that we remembered 10 43 means one 10 4 3 why? Because 10 is less < 4² True and 4 is less than four 10 is less < = 4² True 4 < = 3² True But 10 is not less < 3² This is false. Is it clear? Ok.

[00:37:23]Son, this is the thing. You understand carefully. When you have to prove something, you have to do it by taking xy ab.

[00:37:31]Like I did before. When you have to prove something, you have to take xy or ab.

[00:37:37]But when you want to discover or not discover something which is not true, then you will have to tell us for which number it is coming false. To prove, as we proved earlier, that those relations were equal relations. So we took x y there.

[00:37:53]But to discover it you have to take a particular example. Son, did you understand what I said? Whoever told you that is wrong.

[00:38:01]When you want to discover something, you have to pick an example for which it is false. Did you understand what I said?

[00:38:12]This is math, son, math. Do you know what happens in math? Suppose there is no rule and you have 1 lakh elements. No rule is true for 9999. If it is false for one, then we consider it false. Ok? So if it is false for even one person then it is false.

[00:38:32]When you have to discover, like you have to prove that yes, it is not reflexive, then you will have to tell, for example, if a child has passed, then we will have to tell that he has passed in every subject, but when a child has failed, then we will have to tell him in which subject he has failed. Do you understand what I am saying? Yes. Ok. So when we want to disprove this, we have to say that this result does not hold for this number.

[00:38:59]Like 1/2 is not less < = 1/2² means relation of 1/2 does not come with 1/2. One is enough that the relation of 1/2 did not come with 1/2.

[00:39:10]This means that this relation is not reflexive. Now see, one has a relation with four. Why? Because one is less than equal to 4² but four has no relation with one.

[00:39:20]Why? Because four is not less than equal to 1 squared. There for this relation is not symmetric. Now what is transitive? Since 10 < = 4² this means 10 has relation with four. 4 is < = 3² This means four is also related to three.

[00:39:37]But 10 is not less than 3², so 10 does not have a relation with three. There for 10 is related to four. Isn't it? And four is related to three. But the relation of 10 did not come with three. This me this relation is not transitive. When you have to disprove something, son, you have to do it by taking examples.

[00:39:57]In that case you cannot do it by taking AB. Do you understand now, children?

[00:40:23]Now look, this is the question, now when this comes to you then you should remember 25 32 Jasmeet got it, son knows, Jasmeet son got it, address Jasmeet from Patiala, yes good good now look for this 2532 for this 2532 now look, same thing since 1/2 is not less < = now you know what is the relation, the relation of x with y is when x < = y, yes it will be with y, okay, clear, now look, now 1/2 is not less < = 1 / 2, see what is J?

[00:41:15]It is 5. Now what is J? This is a cube of 5, right? Now this is 0, how much is this? It is a cube of 5, right? This is a cube of 5. Now see how much it came?

[00:41:36]25 125 is 0.125. This is 0.125, okay sir. clear? This is not the case now. There is If 1/2 1/2 Does Belong to R Same Question. In the previous example we had written square here. Now I have written cube here. Any doubt? Same question. The following example is also the same. Now look since 1 is less <= 4 but 4 is not less than equal to 1, okay? clear? There for 1 4 belongs to R but 4 1 does belong to R.

[00:42:14]ok clear ok there for the relation R is not symmetric the relation R is not symmetric no doubt yes ok ji here Q will come now next now from the next actually you know what we have to take 25 32 exactly now look carefully since 25 is < = 3 and 3s < = 2 but 25 is not less < = 3 ok clear clear so there for see 25 this is 27 right 27 has come here ok there for 25 3 belongs to R and 3 2 belongs to R but 25 2 does not belong to R there for the relation R is not transitive. There for the relation R is not transitive.

[00:43:17]Ok? clear? Tell me if it hurts. It is the same question.

[00:43:20]In this also we took the same example first.

[00:43:23]1/2 1/2 ones. Then what did we take? Took the 1/4 one. Isn't it? What did you take here? Here. Then we took 25 32. There was a difference here. The first two examples are the same. Ditto beans. The first two examples are same but what did we take in the last example 25 2532 Riya Ramgarhia Riya ok Riya ok ok ok Rishabh you should write see what is it first thing you know what is it there is a problem in these questions that you cannot change its language like it is written na since because employee is but you have to enter everything like this write it like this what do you do write the question like this write these questions like this write exactly like this it is okay sir do not change their language you will definitely get full marks okay clear so whenever it comes to your mind x <= y² then it should come to your mind 10 43 if the question comes to your mind x <= y then it should come to your mind 2532 first two examples are same 1/2 1/2 so second one 1/4 just learn to write it tell me who all of you understood it if you understood it then write that you understood it yes Sir, write yes sir, if you understood then write yes sir son, this is a question of four marks. Four marks for four marks, yes, okay son.

[00:45:20]10 32 is also fine. 1032 is also fine son.

[00:45:39]Harman Singh ok son 1032 is also fine I have said it son 10 < <= 3q is fine and 3 < < = 2 that is also fine so what is next 10 is not less <= 2 that is fine 10 32 is also fine what is the problem Harman where are you from son Harman where are you from son Kabir I said no right now to say present you will do it only when I say so ok sir I did not say right now for present you have just come ok sir now look for this take its screenshot son and put it take its screenshot for this you have to remember 104 well done you take its screenshot Mansa ok ok son your name is not Haram son take a screenshot of this also.

[00:47:04]Take this also.

[00:47:16]Rohan, there is a test tomorrow, do it well. See, the day you do well, you will get praise. The day you do n't do anything bad, then you know what you get, we cannot tell that to anyone online.

[00:47:29]Okay Rohan son, you know that the day you do well, you get praise.

[00:47:33]Ok? The day you do something bad, you will still get praise but it will be in a different way. Isn't it? A little bit at all.

[00:47:55]Abhay yes you will also face the same situation don't worry okay so now see this is that L1 L2 since L1 is perpendicular to the line L2 defines the set on all lines. Ok? Do you know what? Now what is it to see? Have we proved this? Perpendicular. Perpendicular.

[00:48:34]Ok? So what do you know? Now look carefully. Do we actually have to tell about this relation? What is? One, understand carefully that this relation is only and only symmetric. It is neither reflexive nor symmetric.

[00:48:50]What is a relation? That when will this relationship between L1 and L2 come, son? When will your L1 be? Perpendicular to L2. What will happen?

[00:49:03]What will happen to L2? Perpendicular.

[00:49:07]Ok.

[00:49:21]Now see what is the first one? Since no line is perpendicular to itself. Wow!

[00:49:38]There for L1 L1 does not belong to R there the relation R is not reflexive the relation R is not reflexive because do you know what reflex actually is? OK, is it clear now? It is a very simple thing that it will be reflexive only when every line has a relation with itself. Now this line, this line cannot have any relation with this line. Because that will happen only when this line which I am telling you, when this line is your own? Perpendicular: Any line can be parallel to itself but it can never be perpendicular to itself. Did you understand what I said? So there for this line is not perpendicular to itself. So what did you say there? The relation R is not reflexive. Actually, no line can ever be perpendicular to itself.

[00:50:56]That is parallel. So there for this relation is not reflexive. Isn't it?

[00:51:01]Is it clear? Any doubt?

[00:51:20]Number two.

[00:51:29]No son, it is okay, this is okay, this is so okay, there is no load, no need to tell, let L1 L2 belongs to R, this implies L1 is perpendicular to L2, it is clear. Now this is your L1.

[00:51:58]This is your L2. Has this employee also come to L2? L2 is perpendicular to L1, so this implies L2 belongs to R.

[00:52:25]So there for L1 L2 belongs to R, implies L2 L1 belongs to R, so there for the relation R is R is symmetric.

[00:52:51]ok son any doubt where did number three go let L1 L2 belongs to R and and L2 L3 belongs to R this imply L1 is perpendicular to L2 and L2 L3 belongs to R now look carefully L1 here L2 here your L3 ok ji so what has come this imply L1 is parallel to L3 L1 is what has come parallel to L3 so this imply L1 L3 does not belong to R so there for L1 L2 belongs to R and L2 L3 belongs to R but L1 L3 does not belong to R there for the relation R is not what say ji there for the relation R is not the relation R is not transitive not transitive ok clear any doubt what does this mean this is not reflexive it is not reflexive it is symmetric but it is not transitive also. is not even transitive. ok sir. If it comes wherever it comes, if it comes wherever it comes parallel, what should it come?

[00:55:36]Parallel.

[00:55:37]What should come? Parallel. So son, that's everything.

[00:55:40]Understand one thing carefully. If Parallel comes then that is everything. Everything is there. Meaning it is reflexive, symmetric and transitive as well. But if it comes perpendicularly then what is it only and only?

[00:55:57]is symmetric. There is nothing else. Neither is it reflexive nor is it transitive, son.

[00:56:03]Only what is it? What is one and only?

[00:56:05]is symmetric.

[00:56:16]Now this question is, this question is also very -very related. Ok? So this is not an exclusion, son. This is not. Now see if the lines are parallel. There is a straight Saab in parallel.

[00:56:28]Since each line is parallel to itself. There for L1 L1 belongs to R for all L1 belongs to L relation is reflexive let L1 L2 belongs to R this implies what L1 said is parallel to L2 this implies what L2 said is parallel to L1 it is clear this implies what L2 L1 belongs to R there for the relation R is what R said is reflexive now transitive let L1 L2 belongs to R and L2 L3 L3 belongs to R like this. Ok? So what did this employee say? L1 is parallel to L2 and L2 is parallel to L3. This implies L1 is parallel to L3. So what are you saying there for? Yes. There for L1 L2 belongs to R and L2 L3 belongs to R implies L1 L3 belongs to R. What is the relation R there, tell me? Transitive. So in the case of parallel, it is also reflexive, it is also symmetric, it is also transitive. So those are the three.

[00:57:30]In that case, that relation is equal.

[00:57:33]But what about when the lines are crossed?

[00:57:35]Be perpendicular. So that's the same thing. That is symmetric. Is it clear? Why? Because look, it is a very simple thing. Since no line is perpendicular to itself. There for the relation R is not what bolog? G.

[00:57:48]Reflexive. Next let L1 L2 belongs to R this implies L1 is perpendicular to L2 so this implies L2 is perpendicular to L1 is clear so this implies L2 L1 belongs to R there for the relation R is symmetric what next? Let L1 L2 belongs to R and L2 L3 belongs to R.

[00:58:13]This implies L1 is perpendicular to L2 and L2 is perpendicular to L3 This implies L1 What is it? This L1 is parallel to L3 so there for L1 L2 belongs to R and L2 L3 belongs to R but L1 L3 does not belong to R there for the relation R is not transitive means in case of parallel they are all three but in case of perpendicular they are only one that is symmetric now tell me who all got the address Jasmeet I got my address thank you so much Prachi Bhatia thank you thank you your words have an impact on us okay it gives motivation Ketan Thakur I understood thank you Ketan I understood Ketan Darshana Devi son where are you from what is your name Darshana Devi whose ID is tell me good Jasmeet your chapter is done in the class you are in the class Jasmeet now see there is a question. Now we know that we have to prove that this is a negative function. What do you want to prove? Reactive function.

[01:00:00]So first we proved it one by one. Thank you so much Jasmeet son. Thank you so much. One one is good for Navjot. Navjot Kaur is fine. Ok ok.

[01:00:17]Navjot Kaur SO SOE OK.

[01:00:19]Ok ok son. Navjot son, okay, from Purra, yes, you are from Hoshiarpur, yes, it is done in class, son, your Navneet, new, Jasmeet, I know, this is fx1 = fx2, apply x1 = x2, for all, this domain of x1 x2 belongs to r, so that is your, what do you say, ah, what do you say, one one, if it is on two, then how is it on two, that is, for all y belongs to r, there exists x belongs to r, that means you take this r, okay, this r, take the y of this axis from here, you get x, okay, so let's say that, what is your f, so what is that function, what is your on two, on two, so now we have to prove this, it is a very simple question, you see how to do it, okay, then we will proceed further, okay, okay, Prachi son, let the given function is, let the given function is, F from R to R, okay, that son, I will do it now, don't worry, in this it will be repeated, your fx = 3x + 5 over two for all x belongs to r now look number one first f is one one it is very simple. But just keep one thing in mind that you should not change the language. Whoever did it is gone.

[01:02:52]Let x1 x2 belong to r. This is the domain of r that I am talking about.

[01:03:02]True that fx1 = fx2 Okay. After this, attendance will be taken. This implies 3 x1 + 52 becomes 3 x2 + 52 = 2. This translates to 3x1 + 5 = 3x2 + 5 to five.

[01:03:40]This implies 3x1 = 3x2 This implies x1 = x2 There for fx1 = fx2 Implies x1 = x2 For all x1 x2 belongs to r It's just a simple thing. There for f is verman there for language you have to keep this. f is one one.

[01:04:25]Then look carefully. What do we take? Let x1 x2 belong to r and let r be the domain of x. What to take? fx1 = fx2 You know fx, that is how much 3x + 52 is, this is how much this employee three, you have to enter all these employees. Ok? 3x1 + 52 = 3x2 + 52 So this employee 3x1 + 5 = 3x2 + 5 So this employee 3x1 = 3x2 Employ x1 = x2 So there for fx1 = fx2 Employ x1 = x2 For all x1 x2 belongs to r There for f is 1 First this one has come, it has come easily Mamta Rani where are you from son ok this is Palak Palak which came from DAV Palak is from DAV Now look this one has come Now look carefully there number two app is on to now look carefully f is 3x + 52 right what is it see f is yours r2r true that fx is 3x + 5 Palak. Ok. Ok. Ok. Let y belongs to R be any element put ok ok ok ok ok Payal put ok ok ok ok Payal son put fx = y There for 3x + 5 = y Then There for 3x + 5 = 2y Then There for 3x = 2 / -5 There for x = 2 / -5 3 It belongs to r. Do you know why he came? Why?

[01:07:44]Because your y belongs to r. Look above, what did we write? y belongs to r. Now y belongs to r so that's why x belongs to r. x Belongs to what has come? r has arrived.

[01:08:02]Okay clear so there for there for per all y belongs to r there exists x = 2 / -5 belongs to r true that fx = what is how much is fx? 2 / - 5 3 What do you do now? You should put this where your x is.

[01:08:42]See what will come? Two will definitely come down. Ok? Up three where x what do you put? 2 / - 5 3 + 5 so 3 divided by 3 = 2y -5 + 52. Now look at this carefully, how?

[01:09:15]Look, I will tell you carefully.

[01:09:22]There for it is equal to 3 so it is.

[01:09:26]Wherever your x is, you have to put this 2 / - 5 + 5 over 2. You see, if x is there then it comes 3x + 52. If this is there then it comes 3 * this + 52. Three goes to three. Now what came, it is = 3y - 5 + 52.

[01:09:53]Not gone, 2y, so there, there for 2y over 2 and it is = y. So what did you say for there? So there for f is on two. So there for f is both one, one and on two. What has come? Yes, what has come? Say it, well done. There for there for f is both one, one and on two. Now you have to do this question like this also. We will do the rest later like he took this you take this one what else take 3x - 6 7 is there one you take this what do you say 7 - 3x 5 what is the problem isn't it society has and what is Rohan writing society has less is amazing okay okay so now okay son so now it is done now you do this okay right now we have time only and only till 8:23 right okay so now you recharge your phone okay first eat something have dinner recharge your phone revise the questions that you have asked then we will meet you again at 10 am at 10:00 PM we will meet at 10:00 only half an hour and that's it okay clear right now our class will be held right now it took 1 hour 10 minutes right so we will meet you again at 10:00 at the same time then we will also take attendance still many students are absent I know okay clear so we will not leave them tomorrow under any circumstances okay Yes, he will not survive, otherwise son, we will meet at 10:00 am, the link will be here, we have to answer three-four more questions properly, thank you so much. Then don't say that Vijay Sir did not tell you. If you fail the test tomorrow, we will present the same test before your parents on the last 30th during the PTM.

[01:12:26]Ok? So, for those who will not come, we will send the PDF to their parents' homes. Ok? So thank you so much son. Bye-B, now you look and read. Ok? So son, whoever is watching, please like the video also. Thank you so much.

[01:12:39]Now we will meet you at exactly 10:00.

[01:12:42]We will not get derivatives done.

[01:12:44]I have sent you lectures on derivatives. You do that yourself. We will not get that done.

[01:12:48]I will definitely put that in. Thank you so much.

[01:12:51]Now we will meet you at exactly 10:00.

[01:13:28]Yes children once again welcome to my channel Be Sharma Math Lecture children how are you let's start the game without wasting time okay son first we will do this there are two questions right here on your screen son the sound is coming see the sound is coming son let's start first we will do these two questions the sound is coming so these are two questions. Ok? One is the same as before.

[01:14:15]Look, do you know what? This is linear form. This is linear. Meaning if it is 1° then a line is formed.

[01:14:28]Ok? So this is your upturned parabola.

[01:14:33]Ok? What do we have to tell whether this is one forest or not? Ok? Son, keep one thing in mind when your function is R to R, whether it is R2 R or linear, it is also one one, one one, both are end on two, one one on two, but if the function is R2 R, okay, it is quadratic, like this, type of 1 + X, okay, it is clear, so son, neither it is one one, it is not one one, it is not end on two. Neither is there. It's fine now? When we prove it, okay? So son, we will do it by taking x y.

[01:15:17]But when we disprove, we will do it by taking examples. I am telling you again.

[01:15:21]When we prove it, son, we will do it by taking x and y.

[01:15:24]But when we disprove, we will do it only through examples. Ok?

[01:15:29]Any doubt?

[01:15:33]Son, it will take 10 days. It should not happen that you are out of class in the morning. ok sir. So this is your graph of fx = 3 - 4x. This is the graph. This is the line. Whose graph is this? f(x) = or y = 3 - 4x. This is his graph.

[01:15:54]Correct? clear? So you will see that every portion of it will be covered. First thing. Ok? Will you see it? Tell me sir. If you take different points on the x axis, you will get different ones. What should I say, let's assume its value is x1. This is your x2 so this is your fx1. This is your fx2. The images are coming out different at different points.

[01:16:18]What do you say along with it? Your entire y will be covered.

[01:16:20]This x, this is your x', this is your y, so this is your y', everything will be covered. So this function is 1 and also onto. But look at this, this is your graph, fx = means y =, let's say it is the same thing, 1 + x², okay sir. So you know that whenever we take a line which is parallel to the x axis. If that line cuts the graph at two or more points. If the function cuts at two or more points then it is never one. One is never one.

[01:17:00]First thing. The second thing is that only your entire bye from the top till the bottom was covered, your entire bye was saved. Did he survive? What does this mean?

[01:17:14]That its range is upward.

[01:17:16]What is its range? You see, from where it starts, from one to one + infinity, okay, which is not equal to its code domain, so this function is not on two, it is not on two, this is its graph, okay, if any graph, then I am telling you that if any line which is parallel to the x axis, if it cuts that graph at two or more points, then that function is never one. There is never a forest. Ok?

[01:17:52]All the children came to know about this.

[01:17:55]How to do it now? You look carefully.

[01:17:58]What is your first question?

[01:18:08]Solution Let prom R2 be a function defined as fx = 3 - 4x but all x belongs to r ok clear there for number f is one F is one let x1 x2 belongs to r true that fx1 = fx2 ok? This implies 3 - 4x1 = 3 - 4x2 This implies - 4x1 = - 4x2 This implies x1 = x2 So there for there for fx1 = fx2 Implies x1 = x2 for all x1 x2 belongs to r There for f is one There for f is one This is proved. It was a simple question.

[01:20:15]No problem, yes Jasmeet F Patiala, are you understanding son?

[01:20:21]Divyansh Manoj Thakur, Saurabh Thakur, Mohit, Pramod where is Palak, and Priyanka Rohan Bhardwaj, Rohan very good. I am happy to see you Manas. Ok. Ok.

[01:20:49]Number to let y belongs to r be any element put fx = y there for 3 - 4x = y then there for - 4x = y - 3 there for x = y - 3 - 4 there you have it 3 - y and 4 is the same thing j belongs to R. You will have to write the reason for this.

[01:21:42]Why? Because Y belongs to R, we had taken it above. Yes, look, we took it upstairs. y belongs to R so x belongs to R also. Palak Chaudhary is also there.

[01:21:57]Ok. Ok. Palak hum ok. So there for there for for all y belongs to r there exists x = 3 - y and 4 belongs to r true that fx = f 3 - y and 4 = f how much is it son this is 3 - 4 x what is your 3 - y went from 4 this came out to be 3 - 3 + y and it is = y so there for f is on to there for there for F is on there for what did you say ji F is both one one and on function any doubt means what happened one on means both the negative function has come okay there for f is both one one and on two function function ok any doubt ok it is precious precious Ratika ok now see the next question this is the question this is neither one one nor on two we will describe it by taking an example when we have to describe it then we will describe it by taking an example okay yes simple question Is. No problem.

[01:24:22]There for sus let f from R to R be a function defined as fx = 1 + x not okay for all x belongs to r number one came first since f1 how much? 1 + 1² it = 2 and f - 1 + -1² 1 + 1 it is equal to there for f1 = f-1 see f1 is also your 2 and f -1 is also your 2. But is 1 not equal to -1?

[01:26:00]What happened at two different points? -Your image on both 1 and 1 is coming as 2. What's coming? Two. Meaning there are two different elements.

[01:26:09]But what is his image? It is beans.

[01:26:12]What is an image? It is beans. If the image is the same on different elements then that function is never one. There for but there for f is not one one first thing is f is not one one okay number two since x² >= 0 for all x belongs to r look carefully.

[01:27:26]Number two sin f sorry sin x² >= 0 for all x belongs to r so there for 1 + x² what will it come to? >=1 will come. f all x belongs to r so there for this is your fx.

[01:27:56]Actually this means that the value of this function will never be less than one. The forest or the forest will be bigger. So what are you saying there for?

[01:28:08]What do you say, sir? The range of f, which is the range of the function that is equal to 1 comma + infinity, will come. The value is the range. What's the problem? Is it clear?

[01:28:21]So what is there for range of f not equal to r r? What is r? The domain of f is the domain of f. Whenever the range of the function is not equal to the domain, then the function is never onto. So there for there for F is not on two so this is one one is not on two there for F is not verman not on two okay there for F is not on two now look now we know what are two more questions that I am just giving you a hint there is a function f r to r true that fx = x - 4 for all x belongs to r one this is so what is the second address? f from r to r is true that fx = x - 6 for all x belongs to r. These two are also nothing. Nothing is there Neither is it one on one nor is it two on two. Both are nothing. Neither is it one on one nor is it on two. Ok?

[01:30:12]Now look at solution number one f from r to r is true that fx = x - 4.

[01:30:38]First you have to find its number one.

[01:30:41]You do this, you can take x - 4 or anything. I took Maan Lo To.

[01:30:47]Take anything anyone. Ok?

[01:30:50]Just take zero and you can take whatever else you want. I took it too.

[01:30:52]So what will come there for? x - 4 will come to + - 2. I am telling you the trick. There is a very simple trick for this. 4 + - 2 so there for if you take plus then 6 if you take minus then 2 these are the two values. First take since f2 here, how much did it come first? 2 - 4 it = -2 end it = 2 What's the result? Two.

[01:31:26]Ok?

[01:31:28]Next f6 how much has come 6 - 4 mod it is = 2 mod it is = 2 so do you know what happened there? f2 f6 both are two, it is clear but they are equal 2 = 6 2 = 6 I have taken out two points like this one to one six how much is the image on both, what is two, how much is the image on both, both are not equal, both are not equal but the image on them is same, the function f is not one one there f note one, okay son, okay then do the next om two on two, you know, okay yes number two number two sin fx equal, what has come?

[01:33:12]Number two since fx = x - 4 mod >= 0 for all x belongs to r There for a range of f how much will it come from 0 to + infinity So there for range of f is not equal r r what is the domain of f So there for f is not on to not on to not on to Garima got it, tell me son Garima Noorin Ratika Prachi Ketan Nikhil Kartik second number two this is the last let f from R to be defined is fx = x - 6 for all x belongs to r There for number one first take f greater than this you first take 6.1 what take 6.1 how much did that come 6.1 - 6 it is = 0.1 and it is = 0 what did it come? 0 What's next f 6.2? 6.2 - 6 it is = 0.2 and it is = 0 there for f6.1 = f6 2 but 6.1 note = 6.2 so there for f is not one one there for f is not one one that one is not one.

[01:36:36]Okay son, any doubt? What is its range now? Range Tell me its range. Who knows what its range will be? Hurry up What is the range of this function?

[01:36:49]What is the range of this function?

[01:36:56]Gir bolo what is its range?

[01:37:11]Good Good Nikhil Rana Well done Set of integers Very good What is its range? Set of integers, absolutely, absolutely, well done Garima, now you have found out, good, good set, Prachi's Prachi is visible, where is Prachi, where is Prachi Bhatia, its number will also be found out tomorrow, number two since range of f, set of integers, set of integers, so what is that, not equal to r, but is r code domain of f, code domain of f, so there f is not on two, f is not on two, Prachi Bhati came next, Prachi f is not on two, these were simple questions, so son, I think it is clear, you have found out, okay, first of all, look at this function, you can take anything, what is the trick in it, the question which is about mode, what do you have to do in the question which is about mode, you can put anything in it, xy 4 mode, two put, three put, four put, just a few put, but from there you will get to know what he said, yes, clearly, so do you know what, yes, what is it, put something, your answer will come there You will get points from this, which you have to put only that, this is the question of greatest integer, what you have to do in this, you can take any number greater than six, take 6.1 or 7.1, take anything, take 7.1 and 7.2, there is no problem. ok sir.

[01:39:16]Okay, I think it's clear son.

[01:39:18]So it became ours. Okay, then you will take this and go.

[01:39:27]Mukesh, what is your name son? The Mukesh ID you are talking to is of Master Creder. Son, we will start that too soon. Ok?

[01:39:36]I will tell you the tips and tricks of each chapter.

[01:39:38]Tell me your name, where have you come from? Ok? So now you know what to do and look at the modulus one, the range of the modulus one is non-negative numbers from zero to infinity.

[01:39:55]What is your range for the greatest integer? Set of integers. Ok?

[01:39:57]Whenever 'K is equal to' does not appear in your range code, then understand what it means? Yes, that function of yours is not on. You should practice this. It is a simple question. Is there anything in these? Tell me something. Ok?

[01:40:08]Now we will take your attendance. ok sir. So what did the attendance say? You will get only 1 minute for attendance. So start taking notes, kids. Whom I had asked to note the attendance. Now whoever is absent tomorrow will be thrown out of the class. Ok? So the day after tomorrow, ok the day after tomorrow, son Mukesh, where are you from son? So the day after tomorrow we will inform their parents about the complete attendance. What did you do? ok sir. So thank you so much. Your time starts now. Mark your attendance, son.

[01:41:31]Amritpal Singh Kaladi Okay okay okay son time is over. Now do you know what?

[01:41:40]What should we do? Listen to me. Tomorrow, no, tomorrow is Saturday. Let's see tomorrow, listen to me.

[01:41:47]Very varied lecture which will be held tomorrow.

[01:41:50]Ok? Tomorrow we will ensure at least five to six of your marks. ok sir?

[01:41:55]What will you do? First of all MCQs of Relation and Function.

[01:42:00]One. Ok? Second MCQs of Inverse Trigmatic Function.

[01:42:07]MCQs of Third Matrix. Fourth MCQs of Determinants. Meaning we will do MCQs of the first four chapters. Will do it well.

[01:42:15]Ok?

[01:42:17]We will clear each question one by one tomorrow. It's okay, son. Yes tomorrow is, yes tomorrow is Friday son. Absolutely absolutely. ok sir.

[01:42:23]Tomorrow is Friday. We will do it tomorrow.

[01:42:25]Absolutely, son. So do you know what?

[01:42:29]Tell me what will we do tomorrow? MCQs of G a Relation and Function.

[01:42:32]Second, MCQs of Inverse Trigonometric Function, third, MCQs of Matrix, fourth, MCQs of Determinants and we will do it very well one by one and tomorrow you can understand that you will get four marks out of those MCQs and you are sure to get five to six marks. Ok? He wo n't come out of it. Thank you so much son.

[01:42:55]Bye-by attendance has been noted. Tomorrow we will tell them in detail. Thank you so much son. Bye-by, you keep reading and one should know what to do, when you first listen to the lecture, son, like the video and share it with your friends and later comment and tell me by commenting, I will not comment here, tell me by commenting how you liked the lecture.

[01:43:18]Thank you so much son. Bye-B, you keep studying and keep trying because those who try never lose. ok son bye