 Sets relations and functions. So let me start with the first part, which is sets Okay You all have done sets before right? This is not a new chapter to us Correct. What do you understand from a set anybody? Jacob Prisha Tanishka Veda, what do you understand from a set? Collection of things absolutely basically set is a well-defined This term is very important. It's a well-defined collection or Aggregate of objects collection slash aggregate aggregate of objects Okay Which we call as the elements of the sets Okay, so objects are called as the elements or members of the set Elements or the members of the sets Okay, now this word well-defined is very important. What do you mean by well-defined? What do you send from well-defined any idea? What are the meaning of well-defined? Sorry, I didn't hear you. What was it? Explicit data. Okay, so it should be clear without any doubt that Particular members should fall to that set or not for example so far set right They normally come in a set of three and two two single single right But they are designed in such a way with respect to their color of fabric and I know Construction that you can figure out that this particular set is a part of this so-called set Okay, or probably cutlery set right you can easily figure out that this spoon or this plate or this cup comes from this Cutlery set and not the other one Okay, if there is any ambiguity Then probably we will not call this as a set Okay, now let me give you some examples where you may feel that there is an ambiguity for example If I say set of all intelligent students in your class Will you call this as a set? Will you call this as a set? No, no because there's an ambiguity. What what what makes them intelligent, right? Intelligent is a very subjective term right or a set of all you can say a dangerous animals Now what do you call dangerous animals even sometimes your pet can also attack right sometimes even dogs But dogs can also attack but pet cats may also bite So dangerous is again a very very subjective term right, but something like this set of all states in India It's well defined right you cannot and there's no ambiguity about a particular state being a state or not a state Okay, so it is completely well defined Okay, but in case where you have you know some ambiguity Let's say set of all planets where life exists So there's an ambiguity in that because we don't know whether there are some planets or Whether you know the planets you have included in your list definitely have life in them or not unless until it has been established Okay, so giving you some examples of sets Let me say a Collection of all vowels right so collection of all vowels in English in English. So this is a set Okay, you can say All colors in rainbow Collection of all colors in rainbow. That's a set Okay, so you may get us in a question in your school as one marker Which of the following is not a set which of the following is a set like that kind of kind of questions can come in your exam. I Can also give you like roots of the equation X square minus 2x minus 3 equal to 0 Okay, so you have two roots coming out of it, which is 3 and minus 1 so you can have a set out of that also Okay, so these are some examples of well-defined. You can say well-defined Collection of objects. So at the same time, let me give you some examples of not well-defined collection, so Example of Not a set Not a set You can say Collection of all big cities in India Now, how do you define a big? You should mention a you know particular area. Okay big is a very, very subjective term Okay, or renowned scientist Renowned scientists of this world Okay, what is the noun? Renowned is not well-defined. Okay, so how do you define a renown? Right famous one or an infamous one. Okay Famous one or an infamous one. Okay, so we cannot claim this to be said because these terms that you see they make it Very ambiguous. They're very doubtful. We cannot talk about them being very well-defined You can say all beautiful all beautiful You know beauty pageants Okay Okay, so What is beautiful? What is not beautiful is all in the eyes of the beholder, right? So we cannot know claims as things to be defined Okay, so these are examples of collection of objects which cannot be called as a set which cannot be called as a set Any questions on this? Now coming to the representation of set Coming to the representation of set. So please Make sure the session is very interactive. Okay. Now you're not in a class of 2025 students. You're only in the class of four four students So make sure you are, you know communicating with me No, you can stop typing. You can just speak speak about it to me. So no need to You know hold yourself back from asking any questions normally a Set is denoted by a capital letter Okay, a b cd is what we use for writing a set. Okay, so this is set is named as a capital letter Okay, so name of a set is written with a capital letter Please do not write small letters while naming up while naming a set Okay, and yes, there are two forms in which we represent it So we represent a set in two forms one is called the tabular form Okay, or we also call it as the roaster form Okay, a roaster form Just like we make roasters right name of the people who will be doing a certain kind of work So that is called the roaster form. Okay, we'll talk about it in some time The other form is basically called the set builder form Set builder form Set builder form Also called as the The rule form where you mention a set by no defining the characteristic of its elements Okay, or defining the rules under which those elements are generated Okay, so that will be covered under set builder form. So let me give you an idea of each one of them separately What is a tabular form tabular form or a roaster form is where you List out the elements where you list out the elements of The given set okay within curly brackets But remember this listing has to be done only once for each element Okay, for example, for example, let's say if I ask you write down the Write down the Alphabet's or write down the set of alphabets that you have in the word mathematics. Okay, so when you have a word mathematics and We are expressing it in the form of a rooster form or a tabular form. We will mention it like this M see how many m's are there that two m's right, but we'll only mention it once In a set every element is mentioned only once Right, please do not mention it more than once Okay, so even though we have two a's will only mention one a okay, even though we have two t's will only mention one t So there's one h one e one i one c one s So if I say give me the set of alphabets in the word mathematics This is how you list them down and this is called Representing them as a roaster form. Okay or tabular form Another important thing about roaster form is you may switch the position of the elements also You may write them in any order you want them, but don't mention them more than once So this can be written in any order. Let me write it down and written in any order Any Okay, so while you're listing down the elements or members of a set in a roaster form or tabular form Please mention them only once and they can be written in any order While you're mentioning them in set builder form. So first of all, let me you know discuss. Why do we need a set builder form? Wasn't the roaster form sufficient enough? Why did you require a set builder form? See many a times you may not be able to list down the elements of a set completely Okay, and you may need to tell the you know person who is looking at a set What is the pattern under which you are generating those elements? Under that you need to use some kind of a formula or a rule to represent those elements and Therefore the set builder form becomes very important Even in the subsequent chapter of relations, you would be requiring set builder form to basically express the relationship between two different variables Okay, remember you were doing functions in the bridge course, right? So why NX were variables which were related to each other by some relation So we cannot mention every point on that particular relation. We can mention it through a pattern or through a rule Okay, wherever listing becomes you can say Non-feasible or impractical will have to use our set builder form to represent a set so in a set builder form you represent a set by Using this kind of a expression it contains all X such that X has a Property px has a property px Okay, let me give you some example of this for example, if I say Give me a set a which contains all X such that Such that X is n square plus n Where n is a natural number Where n is a natural number? Okay from 1 to 3 From 1 to 3 Okay, so this is an example of a set builder form where you're not listing down the element directly But instead told a rule to the person who is just looking at your given set That this is the rule under which you are developing those elements. So I may ask you to convert this to a roster form Okay, so if you convert this to a roster form Okay, how would you write it simple take one first feed it over here one square plus one what it'll be It will be two Okay, put then two two square plus two that will be six Okay, then put a three three square plus three that will be 12. Okay, so this is the Roaster form and this is the set builder form. This is the roster form and this is the set builder form So one type of question you can expect In your school exam to come is that they may ask you to write Roaster to set builder set builder to roster also Okay, now many people ask me this question sir if they have given in those two form and they ask me a set builder form How am I sure that they have written the set in a particular pattern for example now two six twelve They have written in a particular pattern because two came from one six came from two twelve came from three What is the shuffle it and give you so don't worry about that They'll always list down the elements of the set in such a way that looking at it You will be able to find the pattern out Okay, for example Let's take an exercise on this. Let's take a simple exercise on this So now I'm going to give you certain questions where you need to Convert one form to the other. Okay, very simple exercise will take let's say write down the following So hope you are sitting with your notebook and pen write down the following in Roaster form Write down the following in Roaster form This chapter is very easy you will find it super super easy Let's say there's a set Okay, which contains all such elements t such that now this symbol is for such that Okay, many books will also use a dash for it simple state line So don't get confused. Don't think it is a division symbol division symbol is slightly oblique Okay, so such that can be either be mentioned as colon or it could be mentioned as a straight line Okay, so D contains all elements t such that t cube is equal to t and T belongs to real number. I think you were already introduced to the Symbol for real numbers in our bridge course real numbers. Normally we write it with a double stroke r Why double stroke? Why not a single stroke? Because normally single stroke r is used for the word relations Okay, remember the name of the chapter itself sets relations and functions So in relations, we'll use a single stroke r in real numbers. We'll use a double stroke. Yes So please respond to this everybody Tell me what will be the Roaster form Roaster form for this T cube is equal to T what elements will T contain? Okay, what else only one Eight, how come eight cube is eight It's like solving this equation Yeah, so they have indirectly asked you they've indirectly asked you to solve this equation So, let me just sorry. Let me just write it in the corner over here So they have asked you to solve T cube is equal to T Okay, now one habit I have seen in students Canceling out terms. Okay. Remember in the bridge course. I had explicitly mentioned this Do not cancel out factors from both the sides unless until you are sure it is zero Okay, so how do you solve this in a proper way? Just keep terms on one side like this Let's keep the terms on one side like this Okay, Jacob that answer is not complete Manishka also. Yeah, so When we take common over here, we get this right, so this means this means They are Three possibilities either T could be zero or T square is equal to one if T square is equal to one that means T could be plus or minus one both Okay, so this set will contain three elements Zero one and minus one So you can say this is your set builder form. This is your roster form for the same set Is the idea clear? Okay, let me take one more question on this Write down the roster form for this Set E Which contains all elements x such that x to the power 4 minus 5x square plus 6 is equal to 0 Okay, and and your x is a rational number X is a rational number Have you been sense rationally rational? I think you have already done that in class 10 Rational numbers are those real numbers which can be expressed as P by Q form where P and Q are integers and Q is not equal to 0 or It can be expressed as a Terminating decimal or a non-terminating recurring decimal Right just recalling your rational number definition of class 10. Yes Tell me the roster form for this So this is a symbol for such that So colon is one and straight line is such that okay, then ishka Let's wait for others to answer And one more confusion about these classes are dear students. Let me tell you it will only happen in an online mode Okay Physical class as of now. I don't know. I'm not sure about if it'll if it can happen even if the lockdown is over because See online is always convenient if you are focused Okay, if you are a person who is distracted then online is not a you know mode of learning for you But I'm sure it's a very small group of four of you for little go all the way to five or six So online will be much faster Right and we can also, you know decide in a different timings if at all you have any one of you or most of you have some issues, so it is always Customizable according to your needs Okay, so let's solve this question Again in this question. It is as good as solving this equation itself Okay, so let me take it in the side So for the second question So x to the power four minus five x square plus six equal to zero We all know that if I take x square as y Okay, it is just like a quadratic y square minus five y plus six equal to zero Okay, we know it's factorizable if I'm not mistaken. It is factorizable as two and three Isn't it any questions? Okay, so that means y could be two or y could be three So from this quadratic two roots can come out y equal to two and y equal to three Now when you say y equal to two it means x square is equal to two Okay, and when you say x square is equal to two x could be plus or minus root two X could be plus or minus root two, but this does not belong to rational Please note that these are certs Root two or minus root two they belong to irrational numbers. So these are certs certs are irrational numbers What is the third? So it is basically Basically an operation where you are praising a number to a power which is not exactly a whole number For example root two root three root five root six These are all certs root nine is not a third root nine is a three Okay So even when you have a three you don't get a rational number from this because this will give you x as Plus minus root three which is which does not belong to rational number Okay, so the answer to this is this set will remain an empty set So we'll talk about it in some time. This will remain a empty set So you just write curly brackets without you know writing any element between or you may also refer it to a symbol like this This is a symbol for null set. Okay. We'll talk about them in some time. This is called null set. Okay This is fine any question so far Hope the pace of the class is also suitable. Hope I'm not going too fast Is it fine It's all fine, sir. All fine. Veda Jacob. I haven't I haven't heard of heard from you yet. I want to hear your voice Jacob myself. Yeah Jacob Jacob used to I didn't online in this thing. No, uh, when you in class 10 I used to sir. I used to I remember Pratik sir holding your holding a mobile phone or Now I'll give you some questions where you have to convert from roster form to set builder form. Okay So write the following in set builder form Write the following in set builder form following in set builder form so Simple one. I think by looking at it only you can come to know what is the pattern the pattern the question is The set is one one fourth one ninth one sixteen One by 25 Okay Now let me tell you there may be various ways in which you can express it Okay Don't be like Very particular about this is the way only that I can write it. Don't look at the answer of your friend Thinking that oh his answer is different from mine. You may have your own way of writing it Okay So your teacher will be aware of all those different versions of the answer Okay, sometimes you may feel that prime numbers are also working odd numbers are also working So it all depends upon what strikes you at that point of time. There's nothing like Fixed answer for such kind of a thing. There may be multiple answers for the same concept Okay, so please try this out and Let's see You may speak out if you want anybody Done tanishka Okay, so Anybody would like to share his or her answer with me X says that X is one by x square. No, no, no Okay, see You may write it like this a is a Okay, now many people do this x Such that x is one by n square And n is a natural number line from one to five Okay Absolutely correct. Nothing wrong in this answer. So x such that x is one by n square Okay, n being a natural number n lying for one to five. Okay, this is fine. Another person may do directly this Instead of writing x and all he can directly say one by n square Such that n is a natural number lying between one to five Both are accepted. You don't have to always write an x Okay, so you can directly write down the pattern also You may also write one by x square such that x is a natural number lying between You know one to five Okay, so various ways are there. So you may write x one by x square also directly x square So that x is a natural number lying between lying between one to five Okay, or x is a natural number x is a natural number less than equal to five That is also fine Okay, so so many varieties of the same answer you can write your teacher will not mark it wrong Okay, any one of them will suffice Is this fine? Now here the assumption is we have written it in such a way that the pattern is recognizable Are you getting my point? So this is the question that I normally get from students Sir, how are you sure that they have written it in the same order? Right, okay They may have you know the pattern We would have been something else and they would have shuffled it to get this All those questions come to me but rest for sure Whatever is the order in which they have written down the elements you need to find pattern in them only Well, you would have appeared for NTSC exams, isn't it? NTSC exams under the MAT portion we have these kind of pattern identification Sequence and all so those concepts will be applicable over here. Okay, so let me give you one more Second question Uh, let's say Yeah, let's say A contains 2 by 3 3 by 4 4 by 5 5 by 6 6 by 7 7 by 8 Try this out simple Done easy Should not take you more than 10 seconds also to do this Should we discuss it? Absolutely, absolutely So it is N by you can say it contains all N by N plus one type of elements Where N is a you can say integer whole number Natural number whatever term you want to use N is a let's say I write this time as an integer Okay That integer is also signified by I Okay Where N is In the interval you can say 2 to 7 Or natural number from 2 to 7 anything will work fine Now there are certain ones which are slightly challenging So let's say I give you something like this 1 2 4 7 11 16 Okay How do you do in this case? Now this is slightly on a difficult side Let me see if you are able to crack this 1 2 4 7 11 16 What is the pattern here? Let's see who is able to identify the pattern Okay, then Ishka what is the pattern here? Can you can you write it down? Okay, let me put this chart 5 only to the host Yeah, you need to select my name and then type there Okay You know what is the pattern but you're not able to figure it out How to represent it That's the inner problem that people face every year Okay Now this if you can see Okay, let me just list down these elements So if you see this particular set of numbers The pattern is obvious but we don't know how to actually write it down The pattern is the difference between them As you can see The difference between them is in arithmetic progression Isn't it? I'm assuming everybody here knows arithmetic progression of class 10 APs Anybody who doesn't know arithmetic progression? Everybody knows it Okay Now all of you see here what I'm going to do Let us say Let us say I mean just assume that I'm trying to add them Okay Just assume that I'm trying to add them Okay And this 10 continues And let's say I go to some nth term Okay tn Do you remember your class 10th notation tn represented Your nth term of that particular sequence So let's say this is this This is your tn Okay Let me say that this sum is s Okay As of now Just understand that I'm adding all the terms And I'm adding it till the nth term So if I know the nth term I will be able to use that formula Right I'll be able to use that formula just like I'd use a formula here Okay So my attempt here or my question here or my approach here is to find out The nth term of this sequence So what I'm going to do is All of you listen to this carefully I'm going to write down the same thing one shifted to the right So the same terms I'm just writing one shifted to the right So if I continue doing it The term underneath will be tn-1 And the last term will just You know Come after that So it will just exceed the final point Correct Just subtract it Just subtract it Okay So what is s-s? s-s is 0, right Correct And if there is nothing you can consider as 0 here 0-1 is a 1 2-1 is a 1 4-2 is a 2 7-4 is a 3 11-7 is a 4 5-6 And so on Okay Till you end up getting this term over here And there is a 0 over here you can consider 0-tn will be minus of tn Okay Now let me tell you why I'm doing this Because it will lead to Certain important thing that will help me To solve the question As of now is there any problem in this Any doubt regarding whatever I have done so far I'll repeat the process I wrote down all the numbers And I'm assuming that I'm going till some nth term This is just to Help me to find out the formula of the nth term Then what I did was I added them all I called it as s I wrote the same s One shifted to the right And then I subtracted both of them And this is what I'm sitting Presently at Any questions here No problem? Okay Now let me take a snapshot of this Because I'll be taking it to the next page I don't want to rewrite again The whole thing Let me go to the next page So this was what we had written On the previous page Now all of you please pay attention This term tn I will take it to the other side And here I will have 1 plus 1 plus 2 plus 3 plus 4 Up till tn minus tn minus 1 Now answer this question of mine What do you see here I see An arithmetic progression over here Whose first term is 1 Whose first term is 1 Correct? And whose common difference is also 1 And number of terms How many number of terms are there by the way Who will tell me Think carefully and answer How many number of terms are there Till this step to this step How many terms are there Who will tell me Tanishka Prisha How many terms are there It's undefined See total number of terms from here To here was n So this was n number of terms n terms are here Correct? So I'm asking you If I drop the first guy How many terms will be there n minus 1 terms Yes or no n minus 1 terms will be there Isn't it? Yes Any questions Okay Now do you all know the formula for Sum of n terms of an arithmetic progression What is the formula Who will tell me Unmute yourself and talk n by 2 n by 2 2a Plus n minus 1 into d Wonderful Now let's say you are Facing an arithmetic progression Whose number of terms are n minus 1 First term is 1 So 2 Again n minus 1 will become n minus 2 Correct? Common difference is 1 So what will be the sum now Can I say this 2 and this 2 Will get cancelled off It will be n n minus 1 by 2 Okay Yellow curly bracket We still have to add this 1 So your Tn would be 1 plus n into n minus 1 by 2 Okay So you wanted to know the pattern right The pattern is this This is the pattern Now this is on a Complicated side It may not come to you in your school exams But be prepared for this also Probably they may ask you this After teaching you Sequence and series chapter Because this concept is very much there In your sequence series chapter So probably in your Semester exams of February They may ask you a question like this So keep your Ideas clear about it So you may say that the set a Is made up of 1 plus n n minus 1 by 2 type of elements Such that Your n is the natural number Okay by the way how many terms I had given you 1 2 3 4 5 6 right So n is the natural number Less than equal to 6 Now if you don't trust me you can put The values and check For example if you put 1 over here What do you get When you put n as 1 You get 1 only Isn't it because this term will vanish Correct What do you end when you put 2 When you put 2 You will get 1 plus 2 into 1 by 2 Which is nothing but 2 itself Right as you can see You got these 2 When you put a 3 what do you get Check 1 plus 3 into 2 by 2 So that will be 4 Correct So this is your third term Okay so this is the pattern Which probably will not be very evident to you You need to work out all these things And figure it out Now there are shortcuts to get this But I will not tell them right now to you We will wait for the right chapter to come To discuss this We will also do it by a shortcut later By a shortcut later Any questions so far in conversion From roster form to set builder And set builder to roster I think the idea is pretty clear in your mind Can this question come first? Sorry sorry I am not able to hear you How many marks can this question Come for Okay if you just ask for this question It may come for a 4 marker Okay It may come for a 4 marker not less than that Okay because guessing this is not easy People just looking at it Cannot guess this This requires a bit of working out Normally a 1 marker questions Are those where you can answer with a 1 word Or a 1 line Okay just see the sample paper which I Shared with you, you can also download it from google 1 marker Has to be answered in 1 word or 1 line Okay maximum 2 lines Want this Next we are going to talk about Kines of sets Kines of sets Okay There is one more thing which I wanted To highlight before we go into this This symbol is very important This symbol is A Greek symbol We call this as epsilon Okay It is to be read as This belongs to symbol It represents belongs to Okay this is very important Because many people Are not very clear about this symbol Okay There is one more symbol that you will come across This symbol Okay This symbol is to be read as There exists There exists So if you see this dyslexic E Don't be like thinking like Probably they have written an E in a different way This is a symbol Which is used for their exists Okay So we call this as an existential Quantifier So it is an example of an existential Quantifier Existential quantifier Okay so these are Your terms that we need to just know Okay not a very You know Rocket science term Existential quantifier There is a symbol like this You must be wondering What maths is full of dyslexic people Writing A in a reverse fashion Writing E left to right or something No It is a symbol Which is universally made for For all Okay So if you see this A Which is written in an upside down manner Read it as for all Okay This is also called Universal quantifier Universal quantifier So these are the things that we need to Okay Start applying When we are solving For problems in maths This symbol we all know Implies Okay For certain sets we have also made symbols Like this is real numbers This is for real numbers Okay If you just want to represent positive real number We just put a plus This is for positive real number Positive real number Hope everybody here knows What is real number and all Okay For integers We use sometimes z Many books you will see the z Being written with a double line This is just to Make it different from the Symbol z which is used in complex numbers Okay Many times we use i for it Okay Z is a word which is carried from The Greek word which means Z is a word for counting Okay This is used for integers Okay This means positive integers Positive integers Okay Many times they will also use Such kind of phrase like R They will put a minus some number What does it mean It means all real numbers Excluding this number Okay Curly bracket means all those numbers Will be removed from there For example they will say R minus 1 and 2 What does it mean It means All real numbers But excluding 1 and 2 from there Can you map on Many times they will use this symbol R minus They can use brackets Okay What does it mean It means all real numbers Except all those numbers Which fall between 1 and 2 So if I show this on a real number Line They want you to remove this number So only this part And this part Would be covered under this So 1 and 2 interval is removed So this interval is your 1 to 2 interval that has been Removed Removed So be aware of these symbols I think whole number, natural number You already know What is rational number Q is used for The word quotient It comes from the word quotient Q comes from the word quotient And it stands for rational number It comes from quotient And it stands for rational number Many times for irrational number They will use this Or they will say does not Belong to rational number For example if they say there is a number X Which does not belong to rational number What does it mean They automatically mean Q belongs to Irrational numbers What are irrational numbers What are irrational numbers Numbers which cannot be expressed As a P by Q form Okay As decimals they are Non-terminating, non-recreating in nature For example Pi For example E For example root 2 Root 3, root 5 7 to the power 1 third They are all irrational numbers Let me tell you a funny thing There are more irrational numbers Than rational numbers Okay There are more irrational numbers In fact they are of order Than rational numbers So this is a symbol used for irrational numbers Okay So if I use these symbols please do not get surprised Okay Any questions here Complex number, somebody is asking complex number This is symbol for complex number C with a line in between This is for all complex numbers Complex numbers Okay So I was talking about Kinds of sets So under kinds of sets We will first talk about Null set Also called empty set Empty set is a set which doesn't contain Any elements So we normally represent a null set By just two curly brackets Which doesn't have anything in it Okay Now it doesn't mean zero It's a set which is empty Correct It's your cupboard which doesn't contain anything Correct So your cupboard is an empty set Okay Many times we use the symbol Phi This is to be read as Phi Okay I hope you are aware of those Greek symbols Alpha Beta Gamma Delta What is called? Correct This is called row Okay This is called mu Okay This is called kappa This is called tau This is called phi This is called theta Okay, so these are the symbols that you will be You know coming across in maths Especially theta, alpha, beta and all You will be coming in trigonometry chapter So I will repeat their names If you want me to repeat them This is phi This is tau Okay, you will be using this In your physics chapter When sir is talking about torque For torque he will be using tau This is kappa Okay Alpha, beta, gamma Delta Sigma Sigma This is row This is row This is mu Mu Okay, so we represent A empty set by A Phi symbol Give me some examples of sets which are empty Give me the set builder form Of a set which will not give you anything Remember we had already done an example X to be power 4 Minus 5 X squared plus 6 equal to 0 Where X is a rational number And there was no element in that set So that's an example of a set which is Empty set Can you give me more examples of sets which are empty Any set builder form you tell me Which will give me an answer Which will not contain any element in it I am sure you can think of many Many such examples Just one of you to give me an example Any example Unmute yourself And explain to me a set Such that when I write it in Roster form It doesn't have any element in it X is equal to 2 X is equal to Equal to 2 X equal to 2 X cube is equal to 2 X cube is equal to 2 Okay, so 2 to the power of 1 cube 1 third is an element You want to say some other condition on X Where X is an integer Probably Yes If you just say X cube is equal to 2 There will be an answer for it So your set will contain some element X to the power of 10 is equal to X to the power of 2 X to the power of 10 is equal to X to the power of 2 it will contain No 1 and 0 A set of all X Such that X is between a 1 and 2 Or the 1 and anything And X is a natural number 1 and anything and X is a natural number Yeah, like any number You can put Okay, so you can say X lies between Let's say 1.2 and Let's say 1.8 And X is a natural number Yeah Absolutely correct I can give you so many other examples One other example I can give you is Let's say X contains X is a Value which satisfies 2 X plus Let's say 11 is equal to 3 And X is a natural number Okay, so when you solve this You realize that 2 X plus 11 equal to 3 will give you 2 X as minus 8 So X is minus 4 Which is not Which is not a natural number Which is not a natural number Which is a symbol that you need to Use for saying not belonging to Let me make a Slight change here This means belongs to And this means This means does not belong to By the way, the idea is If you strike something For example, if you strike E That means there does not exist Okay, if you strike this Means it does not imply Okay So use that also whenever You want to explain something Okay, so minus 4 doesn't belong to A natural number so this set will be An empty set This set will be an empty set Okay, any questions here? None Next is singleton set Singleton set What's a singleton set? It's a set which will contain only one element in it Okay So it's a set which has only One element in it And that element may be 0 also It doesn't mean, see many people get confused This is sir Is this a null set? No Even if it has an element 0 You should not call it as a null set Null set means it should not have Anything in it, not even 0 Okay This is a singleton set Singleton set means it has only one element inside it Okay So can you give me some example Of singleton set Give me set builder form Of singleton set For example, set of all Capitals of India Right Set of all, let's say x where x is Capital of India As far as my general knowledge is concerned I know there is only one capital of India Which is New Delhi Okay So this will be a singleton set It will only have New Delhi in it Can you give me some mathematical examples Also Give me a set builder form So that the answer is only one Give me slightly you know Good one Okay So that x square minus 2x Plus 1 is equal to 0 Okay And x belongs to real number You can speak out, don't need to type With a very small batch Yeah, x minus 2 equal to 0 And x is a real number Okay, so you can see that the answer here will only be x equal to 1 Sorry, only x equal to 1 So these are examples of A singleton set Okay Next is Finite set Now in order to finite set In order to understand finite set There is something little prior to that Which is called cardinal number of sets Cardinal Number of sets What is cardinal number of sets Cardinal number of a set Is the number of elements in the set Okay It is represented by N with the set written inside packet Okay For example, if I say Sir, can you redefine it? Yeah, cardinal number of a set is basically I'll write it down over here It gives you the number of elements It gives the number Of elements in that set Elements in the set And it is represented by this symbol So let's say if I have A, E, I, O, U And I say what is the cardinal Number of this set Basically I'm asking you how many Elements are there in this set So you'll say 5 Okay Let me come to finite set What's the finite set Finite set is basically Those sets Whose number of elements Is Fixed or finite You can say that It has a finite value For its cardinal number So cardinal number is a finite value For example, A itself Which I have written here in front of you Set A which is A, E, I, O, U It is a finite set Because the cardinal number Of it is a finite value Getting the point So those are called finite sets Okay Many a times you may write a finite set Like this 2, 4, 6 And dot, dot, dot till let's say You can write it as 100 Okay, this is also a finite set Because even though the number of elements Maybe let's say in the tune of 50 Or 100 or 1000s or millions They are finite in number Okay, so this is a set whose cardinal number You can say is 50 So this is an example of another example Of a finite set Okay But on the other hand If I give you something like this If I give you something like this 2, 4, 6 Dot, dot, dot I don't stop anywhere Then this would become An example of an infinite set Infinite set This would become an example of An infinite set Okay, for an infinite set There is no cardinal number Defined, the cardinal number Will tend to infinity Okay, so Few examples of infinite set That you can claim From your daily use is real numbers Right Real numbers is an infinite set Right, natural numbers Many people will like And with a double Integers Whole numbers They are all infinite set Set of all prime numbers They are infinite set As of now we have not yet concluded that There are limited number of finites So that's an infinite set So for infinite set The cardinal number is not defined It will be Tending to infinity Now there may be There may be infinite sets In terms of intervals So let me talk about that also with you Hope you don't want to copy anything from here Can I just slide To the right side of the board Intervals I am sure you would have been exposed to Intervals to a large extent In our bridge course program So When we are talking about intervals We are really talking about A set of points Whether on real number line Or whether on Natural number Or whether on integers For example, if let's say I say X is a real number And X belongs to 1 to 5 This is an interval Because If you represent this On a real number line You will represent it By all set of points Which lie between 1 and 5 Including 1 And including 5 So yes, this is an example Of an infinite set Because you cannot count the number of elements here There are so many infinite Many real numbers between 1 and 5 So this is a Special type of a set Which we call as intervals We don't call them as set We call them as intervals For a change Anishka has a question So we did this on a graph Where we are plotting a graph for a certain interval of X For a certain interval of X So can you define it again? Yeah, interval is Basically an infinite set Right But still between 2 fixed numbers Okay, so this is an example Where you are having Infinite number of real numbers Between 1 to 5 So we say it's an interval You want a definition Who was that? Anishka, Veda, Prisha Veda, Veda You want a definition of an interval I am not able to hear you Sir, we can't hear you I think the internet was unstable from my side I am extremely sorry What is the definition of it? Sir See, don't understand From definition point of view Because in CVSE we never ask Definition and proofs Just understand it If your set is mentioned As an interval Let's say I write it as an interval 1,5 with bracket What is the meaning of it? The meaning of it is All real numbers which is present Now you may be seeing this as a fixed line But let me tell you in that line There are infinitely many real numbers In fact, we say that between Any 2 real numbers There would be infinitely many real numbers So the intervals are examples of Infinite set Just now we talk about infinite set But infinite sets which are Continuous in nature And as intervals So you can treat this as a definition These you can call it as Continuous infinite sets Okay Now interval also is divided into 2 types One is called close interval And the other will be called as open interval I am sure when we were doing Redefinition of functions We talked about x So we were using something called less than equal to And only less than So let me explain you that also When you write square brackets So let's say I take an example x belongs to 1,5 And you put square brackets Square bracket means This is an interval where This is an interval where 1 and 5 both are included in your Real numbers But if you write something like this x belongs to Round bracket 1 and 5 What does it mean? It means 1 and 5 will not be included So this is what we call as open interval Now you must be asking yourself Where is anything open over here I still see a close bracket It is called open interval Because when we started using these expressions We used this expression for it Like this This also means the same thing This also means the same thing But you will find this notation very rare I have not seen many books using this notation They will use round brackets only So what does round bracket mean? Round bracket means In your real number line You will include all values from 1 to 5 Everything which is between them But not 1 and 5 themselves So we remove that 1 and 5 Either by making a gap like this Or you make holes like this Both are fine Holes means Those values are missing from your Given interval Is this fine? You may have a combination of each one of them also Yes, each bracket has some meaning In maths, please make sure You understand the meaning of these brackets However, for those special functions Which you would have learnt in your bridge course Like GIF function, fractional part function They would mention it specifically That this bracket means this But if you don't mention anything Please note your normal way of interpreting will be applied For example, if I say X square plus X is equal to 0 For all X belonging to 1 to 2 Okay So basically what I am trying to say I am just making a hypothetical scenario Or you can say let's say Greater than 0 Let me use a valid one so that It at least sounds correct So when you are reading this How will you read it? The question setter is saying that This is an inequality which will hold true This symbol means what? For all Correct? Remember the previous page we talked about This symbol For all X which belongs to This symbol is for belongs to Okay And close bracket 1 to 2 means How will you read it? Means all X which is from 1 to 2 Including 1 and 2 as well Okay So this is what the statement is This is how the statement is supposed to be read Are you getting my point? So these are the mathematical symbols Which will be Coming your way when you are dealing with Class 11 maps Now as I was telling That there may be Mix and match of symbols also For example they may say X by X minus 2 is less Than equal to 0 Okay For all X belonging to 1 to 2 What do you How do you read this? All values including 1 But excluding 2 So all values starting from 1 And stopping just before 2 At 1.99999 So in 2 is not included in your interval Okay Prisha I did not understand your question exactly Sir how is each bracket read help means How is each bracket to be read? Okay so you will read this as X belonging to 1,2 Close at 1 Open at 2 This is how you will read it Okay This one you will say it belongs to 1,2 closed If you just say 1,2 close means both Side will be closed If you say 1,5 open That means both side will be open So here you will read 1,2 Close at 1 Open at 2 This is the pronunciation Is this fine? Any questions here? Any questions with respect to Finite set Infinite set Null set Singleton set Any questions here please ask Okay So let me now move on Let me now move on To the concept of Equivalent sets Equivalent sets What is an equivalent set When you say a set A is Equivalent to set B Not a very important concept I don't think so any question will come on this For your information When you say a set is equivalent to another Set B first of all note down the symbol It is a symbol which we use As a you can say Double side arrow just like We use it for a reversible Reaction in chemistry In chemistry you probably use This symbol right Okay so this is The symbol which we use for Equivalent sets so when do we Call two sets as equivalent The definition is If their cardinal number is the same Okay but remember you cannot Use it for an infinite set Okay so you can't set two Infinite sets even though their Number of elements are infinite Each their number of elements are equal I told you in the bridge course two Infinities are not the same So when the number of elements In set A and set B are equal We call them as equivalent sets Simple example I can give you is Let's say A contains 1, 5, 7, 8, 3 Okay and I said B contains A, E, I, O, U Since both of them contains 5, 5 number of elements each They will become equivalent sets Okay nothing big in this Very simple concept Next concept that I want to talk about In the last 20 minutes of the class That's very important Subsets This needs a bit of Understanding because this is Probably important from your school point of view What is a subset? How you define a subset Okay If you say A is a subset Of B Normally we use this in our daily Communication also for example just now I told you in the beginning of the class That applied mathematics is a subset Of the standard mathematics The symbol that we use is A cup kind of a Structure with a dash there In between This is a symbol for saying A is a subset of B Now how do you define a subset? When you say A is a subset of B That means A contains All such elements which also Belong to the set B Okay So when all elements of set A Also belong to that of Set B A will automatically become a subset of B A simple example How could you repeat? If all elements of set A Also belong to set B We say A is a subset of B So this is the definition You can say of a subset So when A is a subset of B This is how we define it A contains all Such that X also belongs to B As an example Let me give you A simple case This is your set B Okay Let's say I write a set like this Then this A will be called As a subset of B Because all elements of A Also are elements of B But not the other way round Right Not all elements of B Are elements of A Okay So A is a subset of B And you can say B is a super set of A B is a super set of A So this is the Terminology that we normally Use B is a super set of A Okay Now what is the symbol for that You may call the symbol to be A is a super set of A like this Okay It's just the same thing written in the reverse order But this is very less used This is more widely used Okay Now my immediate question to you would be How many subsets of B you can form How many subsets of B you can form So how many such sets A you can carve out from B Such that all elements Of A are also Elements of B So the possible number of subsets Let me write it down The possible subsets are Okay You can say one Okay From B I can carve out a set Which only has one in it A singleton set probably Okay It can only have just two in it It can only have just three in it Okay It can only have one and two in it Okay It can only have one three in it It can only have two three in it Right Can I say It can also have All the elements of set B also One two and three Because it is still under this definition So unless this definition is True or violated We can state all the elements also Correct So one two three also can be a Subset of One two three And one more important thing we are missing Is the null set Okay When you don't write anything That also is a subset of any set Okay So null set is basically Subset of All the sets Is a subset of So null set Is a subset of All the sets Including itself Is this fine Now count how many subsets have you written Sorry how many subsets are possible One Two three four Five six seven eight Yes or no Now can anybody of Can anyone of you relate The number of subsets Of elements you have in the Given set B So if set B has three elements The number of subsets Coming out from B are eight in number So can you strike a relationship Between this three and eight Common sense Anyone Sir could you repeat Yeah my question is How is this number of subsets Related to the Cardinal number of set B Two n first one Two n plus one Two n plus one So two n plus one should have Given you the answer as seven In this case but your answer is eight Okay let me take another example Probably from there you will get an idea Let's say Let's say I have a Set X Which has only got five and six Okay how many subsets can you Make from this No Prisha that is also not Let's take more examples Probably the pattern Will be evident from there So yes how many possible subsets How many possible Subsets that's right Five Can I say null set first of all Can I say set with five Can I say set with Six Can I say set with five and six Is there any other possibility Is there any other possibility No So when you had two elements here There are four subsets Coming out of it Now are you able to see the pattern When there were three You got Eight subsets out of it You got eight subsets out of it Now anybody sees the pattern here Now would you like to change your answer Would you like to change your answer Prisha now your n square minus one Fails Okay let me take another example Okay let's say there is a set Called W Which has only got one element Okay how many subsets are possible How many possible subsets are there You will say null And the set with one So you have two subsets Is it two to the power n Right there you go This is what I wanted to give So now for these examples It becomes very evident that Let me write it in a box fashion So that you should never forget this Okay If n is the Cardinal number of a set Cardinal number of a set Okay Then the number of subsets Hope you can understand this symbol It stands for number of subsets From the set will be two to the power Na Okay So whatever is the total number of elements Just raise it to the power of two That will be the total number of subsets Is this fine any questions There will be one question definitely coming in your School exam based on this particular property It's a very important one Please mark it with a star Okay Now Let me finish this over here Out of all the subsets that you have written The set where you had written The Set B itself That is called An improper set Or you can say improper subset Let me write it like this Improper subset Let me say out of these eight subsets Seven of them Are proper subsets And the last one that you have written Is an improper subset So you may call these seven as Proper subsets Proper subsets So what's a proper subset Proper subsets are those subsets Which has got at least one element Less than the original set At least one element Less than a original set That's what I am defining Prisha Proper subsets Are those subsets Which has got At least one element Mark the word At least one element less than the actual subset Actual set That is called a proper subset If your subset is exactly Equal to your given set It will be called as Improper subset So out of Two to the power n subsets One of them will be improper So you can break this up as One improper subset One improper subset And the remaining That is two to the power na-1 As proper subsets So sometimes your teacher may ask you That if a given set has got Five elements How many proper subsets you can make What will your answer be If a given set has Five elements How many proper subsets are there in that set What will your answer be Four If your set has Five elements How many proper subsets are there Nine Nine Just follow this formula Two to the power Five Minus one How much is this 31 Is this fine Any questions here Now remember one thing Let me write this as a Extra point for you Note well Five is an improper subset of itself Why? Because when the set itself is When the set itself is written as a subset It is the improper subset Okay Just like one two three itself when it was Written it was improper subset Isn't it? So when you are just having a null set Then remember null set will be its own subset But that subset will be Improper That's why This particular statement I have written Null set is an Improper subset of itself Please understand this concept Because it is going to be Used in subsequent concepts Like power set and all So Please get this right Right now itself Don't wait for data time to come to ask this We will have more examples Don't worry Let's write down the subsets of Write down the subsets of The set which contains The letters L E O Leo Please do that We just have to mention how many subsets Can be formed or No no no you mention the sets also You mention the subsets also Done? Simple? So what are the subsets? Let's write down the subsets Subsets of A So Null set start with the null set Start with the set having just L E O Then two at a time L E L O E O and L E O Again you can see the count will be 8 only Because the number of elements here are just 3 And this last one will be called as the Improper subset Any questions? Any questions here? Is this fine? Now let me ask you this This is the question which normally Confuses a lot of people So let's say this is your subset This is your set A Which of these two symbol is correct symbol? Both are correct I have given you two Notations Tell me are they both correct? Or only one of them is correct? One of them is correct? Which of them is correct? First one First one is correct Why not second one? Some of you are saying both are correct What do you think? Let me ask Jacob Jacob what do you think? Both are correct or one of them is correct? The first one is correct Oh yeah poll Let's have A and B for it Okay Let's have A and B And this is your B statement Okay By the way there is no response For both of them If you feel both of them you can type it out Okay so two of you think both That's why you have not voted Okay one of you think it's both Okay all right Now here Let me share the result So remember this is a wrong notation The second one is a wrong notation So only first one is correct You can't say A set is a Element of a set Unless until that set Itself involves elements like sets For example in this case L is an element of that set So right way to write it is L is an element of the set But the moment you put this curly bracket Right There is no element like this inside it Can you see any curly bracket L over here No right In this case In the present example Writings B statement would be a blender So this will be wrong So this is the right way to represent it This is a subset you can say of A Are you getting my point This is where people don't realize The difference between The set And difference between element Now if I give you a question like this I'll take another question These are very conceptual questions Your teacher made Ask questions on this If let's say I write it A set which contains L E O Now see what has happened Not only L E O R Elements of this set But a set with L And the set with E O R also elements Are you getting my point Sir can this be considered as a subset No wait Let me ask questions here If I say write down the subsets of this Okay How will you write Let me take a simple example So that you can understand the difference Let me remove out some unwanted things Because it will be very long a problem Let's say I write it like this Okay Now please note I have not repeated Any element These are two elements present in the set So set itself So set itself can be an element of a set Getting my point Right So cupboard may itself contain a cupboard Getting my point So If here if I ask you Is this A subset Of A Or this an element of A You'll say both are correct Both are correct Okay Because You can make a subset out of this element Which will match with this So this can be a subset of A And there is an element like this also Okay so both are correct In this case But let's say if I remove this Let me take a separate example If I say a set contains Only of an element Which is a set itself Correct Then what will you write Will you write this as an Element or a subset of A What will you write Choose will you write this symbol Or will you write this symbol Yes I am still waiting for the response Belongs to a subset Please type it out if you are not able to speak Absurd No my dear It will be belonging to Now this is behaving as an element Let me show you what will be the subsets Subsets will be now set Okay A set containing the Set L which is the set itself Okay Neither of these two is actually this This and this are different Let me tell you These two are different This is a set containing L This is a set containing An element which itself is a set This is where 99% People fail to understand The meaning of a set Are you getting my point So think as if this is a bigger Elmira and this is a smaller Elmira Out of it So if I take out a smaller Elmira and ask you Is this an element Of your bigger Elmira you will say yes Are you getting my point You will not say it is a subset Of it Okay so in this case it will be An element This is not a subset This will be a subset So a set with That element will be a subset of it So these two rotations are correct But here please do not write This is a subset of it No this will be wrong This will be wrong This will not be a subset of it Are you able to get to this point What do I mean by See many people fail To understand that Itself could be a set in itself Okay So when you are representing a subset You have to put double brackets outside it Just like I have done over here Let me take one more question Then only we can close this session One more question Let's say Write the subsets Of Write the Write the Subsets of One Two Let's say TQ Now see here Set A contains three elements This is one element This is another element And this set itself is another element Are you getting my point So think like there is a cupboard Big cupboard A Within that there are two cupboards cupboard B and cupboard C cupboard B has one item inside it cupboard C has got two items inside it And the third item which is two Is outside all the cupboards Okay So it's a situation like this There is a cupboard Within that there is one more cupboard And there is one more bigger cupboard Okay So there is a cupboard which contains one There is a cupboard which contains P and Q And there is an outside element too Okay So this is how basically this particular Set can be perceived So this is an element of the cupboard A This is the element of the cupboard A And this is also an element of the cupboard A Okay So please accept the fact that Sets also can be elements of a Bigger set Now see For example real number Real number itself contains sets of Set of integers itself contains Set of natural numbers Isn't it Okay Now if I ask you to write the subsets What will be the subsets? See the solution Null set The very first subset that you should always write Is the null set Next is now see All of you please watch out here The next set will be a set of One Set of the set one Don't write it just like set one It will be wrong Okay Next would be set of two Next would be Set of two Next would be set of Set of P and Q Correct Next would be set containing one And two Next would be a set containing Two And P and Q set Okay Next would contain the set of One and P and Q And finally will be The whole set itself Which is your improper subset Okay so count How many subsets are there One, two, three Four, five Six, seven Eight Total number should be eight Because there are three elements inside this The power three which is eight Is this clear, any questions here Now you understand Now if I ask you this question Let me put this as a final question Is this an element of A Or is this a subset of A Type it out Yes Concluding questions of the class Is this In this question is one the element Is a set of one is an element of A Or is it a subset of A Jacob has already answered I want everybody to answer All the four of you Tanishka now I'll faint Yeah Do you see this as a subset my dear Hmm This is An element of A Not a subset my dear If I had to write it as a subset I would write it like this This is a subset of A Okay You have already written all your subsets No just check Have you written any subset like this I don't think so If you are confusing it with this Let me tell you it is a different thing This and this are different things Okay Alright so with this we conclude this But one isn't there What is the meaning of one isn't there Is a set of one Not an element of A Look here So that is the answer It is not a subset of A If you had to write it you would have written it like this And this is different These two are different things There is an extra curly bracket outside here There is only one set of Curly brackets Getting it Tanishka Okay so Don't worry I'll be sending this video to you I'll re-watch this video This concept is important Last one don't take it easily that Okay no it's fine Even if I don't understand this no There is a topic called power set There is a subtopic in sets called power set That is coming to you next class When we have Next class would be Of a slightly more duration Two and a half hours is fine Any problem with Two and a half hours It was your first class that's why I thought I would keep it slightly lighter