hello everyone how are you all so let us continue our discussion on States relations and functions last lecture maybe did a very elementary discussion we saw it was said it is nothing but a well-defined collection of objects relation clear hair dos excavation of linckia head pattern established courtesy hum relation gated there and whereas function it is nothing but a special type of relation now what are the special conditions to be imposed in relation we have algebraic it ok so we had clear till this point said Keota hair relation to your function ko kya when is sex by detail discussion gonna start kieth on how many have you said kya hai equal defined collection of objects kahafer elements connected members pair coach elements car it s er collection Jo elements upper spec common characteristic property share cut the hoe who see him said hey hey Casey property come eat them strictly refined car by kiko t sir element who set me up ahead Jeff in the hey up ahead for example on Radhika the reverse of India so up hum defying curse of the heck in nine rivers of India minima here ta but Ganga rivers of India may at the hem this has to be such a belt crisp dissipation of a collection okay so this is a set clear how many representation of sets taqaddum we saw sets can be represented in two forms roaster form and set builder form doliveira Giga representation America day and then we saw one more definition that was number of elements in a set so number of distinct elements in a set is said to be number of elements in a set now divorced continuing further let us solve some questions on what we have studied till now or Skiba we will see different types of sets okay capela couch normal say basics Adobe Theory concept Perl ahead on because questions they click a gel listen then we will discuss about different types of sets okay so let us see some questions first question they came so we have to write the solution of the set of the equation so question is we have to write the solution set of the equation X square plus 2 X minus 2 equals to 0 in frost or form rooster form ma is equation gas solution set looking ahead ok so this is a quadratic equation I notice got going to have two solution or my ng7 liquid Barley Mow keep those solutions here hungry see I will write this as X square and this is X so let the XK plus 2 X minus X minus 2 equals to 0 so this is X plus 2 X minus 1 equals to 0 so it has two solutions minus 2 and 1 a roaster for Malik Mejia other elements cook curly brackets kovitch malignant head or comma separate Carina has this is the solution set for this equation first of all a very easy one next question we have a set which is given by the other set a which is given by 1 4 9 16 25 and so on ok in set builder pump we have to write this set it we have to write this set in set builder form roast up on my head how many said build a formal ignite of MA Enki common property they separately but that’s a common property can see this is 1 squared this is 2 squared this is 3 squared this is 4 square this is 5 square and so on please sorry natural numbers K square here any common property hanging may so homie same set builder for making get X square such that X is a natural no perfect okay next illustration licked ahem so next illustration is we have to write the set 1 upon 2 I would like this set here so let us see we have set B which is of the form 1 upon tune 2 upon 3 3 upon 4 4 0.5 and so on I make a set of a set builder for militia half so set builder for Megan ladies this way this is P upon T plus 1 such that P is a natural number he patented a roaster for mayhem the exactly 1 upon 1 plus 1 2 upon 2 plus 1 3 upon 3 plus 1 4 1 4 plus 1 and so on who is he complicit builder for me is there a really clear now there is one more question miss a theoretic messy

really girl 81 so the question is match each of the set on the left scrape describe the roaster form with the same set on the right side this way below said build upon I think he she’d observing a passive sheet measure question such metallically okay so hemara bus left set me a principal a set with zero element one separate one two three six nine 18 or a k3 -3 so Mary pass HR said sam mcvey be no force it is all the alphabets in the little principle okay when the next set we have is this 0 a 10 1 2 3 6 9 or akin 3 comma minus 3 Omega say right value column case at match cottage on ahem so forces X is a positive integer unless division of 18 so paella set builder for mayhem X is a positive integer in a set of and a division of 18th yay kis kisko stuff on him yeah erm a second Joe hum para hey bookcase chisca said builder for man so very clearly this guy over here it is a positive integer and all the divisors of 18 are present here so this is the set builder form for it this corresponds to a then X is an integer and X square minus 9 is equals to 0 X square minus I is equal 0 means X is going to be the 3 or minus P because it mean integer and this is the cell phone it C is X is an integer X is an integer and X plus 1 is equals to 1 X plus 1 is equal to 1 which means X has to be 0 and this is the solution for it X is a letter of the word principle so ya and observe one thing key principle me because there are two piece we just listed once because roster form a hum sip distinct elements he leaked they have okay now next question we have to state which of the following are sets and justify answers okay so I whichever they is my logic am scoring a how many prove conically a key set thickening we just have to prove that these are well-defined collection of objects such that we can easily identify that one element follows two belongs to this set or not how many of us here the exactly Ethne well-defined collection heck in any chaos on a service Apogee Co element is set me up a hair linking up at me immediately okay so the collection of all months of a year beginning with a letter j collection of all years of the month all month of an ear with beginning with letter j okay we can see i know Jan very it will belong to this set because it is a month of the year and it starts with J February no because it does not sauce with Jane March no April no mail no June yes because it starts with J July yes because it starts with J is 30k some go hot crispy define carmine a key weather a month belongs to this collection or not so this is a set okay second the collection of ten most talented writers of India is this a set or not I will say let us see the collection of ten most talented writers of India AB so GM k other make we writer can Amla so can it crispy say is this the ten among the 10 most 10 most talented writers of India because we cannot define it is always a related to Lichter in fact hum toh magazines has the newspapers the competitors as they will give their own different list of the ten most talented writers this is a relative term I am Kira spinning in a fine car Pengy that weather an ABC writers belongs to the sect or not we cannot so this is not a sec caught it a team of 11 best cricket batsman of the world same we cannot define because this is again a relative term how crystal a divinely he could mine give is a Sachin Tendulkar belongs to this collection or not or whether Mahendra Singh Dhoni or whether Ricky Ponting belongs to this sect or not because because ugly so chilly this is in Iowa created as Arthur shangai PJ collection amateur change with or heckum so we can never crisp redefine so this is not a set a collection of all boys in your class is this I said I say yes because I have a point to a boy you can very well surely say that yeah yeh mera class my hair Yemeni classmen nahi hair so he belongs to the collection he does not belongs to the collection it is a set the collection of all national numbers less than 100 yes again the collection of all natural numbers less

than 100 very obviously again we go 99 through it is a national number less than 100 so it belongs to the collection if I say 113 no because it is greater than 100 or if I say 22.2 so no because it is not a natural number this definition is very crisp oh it is Earth set a collection of novels written by the writer moon ship Ranger yes because other Mako novel Camus for example let us name some novel one witha so this is the written by moonship ancient or not no it is not so this is no so this does not belong to the collection business if I take a novel which is written by moon Cyprian shell so it belongs to this collection so yes it is a set okay the collection of all even integers again yes because making SSTO quickly number integer high any of the even higher we’re even integer hookahs it belongs to the relation or s does not the collection of questions in this chapter yes again because we know whether this is a question which belongs to this chapter and also this is a set a collection of most dangerous animals of the world again not because fit being on shore it is I’m a key element the animal dangerous Hany and it’s again a related term so this is not a sitter so up up of fear operating kiss the rickety collection go home set K they have kidney crisp definition show you before a collection to be a set or not okay now let us see the next question so the next question is we have been given a set a one two three four five six and we have to ensure we have been given some fill in the blanks followed by it or amazement belongs to not belongs to sign Lagaan ahem so it is very easy if they have five you know five belongs to a so I read five belongs to similarly if I have zero I know it does not belongs to a to be light does not belong so you see that it is a buck is already filled cursor is very easy now next question we have been given some section set builder form or remove a roaster form a lick nahan so let us take this one X is a natural number less than six X is a natural number less than six so can we write it here so B would be natural number less than 6 1 2 3 4 is the realism roaster for making it then X to the 2 digit natural number such that the sum of its digits 8 okay let’s see X is a two digit natural number such that the sum of its digits is it toward is it can assalamu says that the sum of its it is it now see how many such numbers we can have I can have 18 no I can have 17 well the sum of the deuces it so we can have 17 we can have 26 we can have 35 here we can have 44 here we can equip tip see here 60 to 70 1 and 18 so these are those to drill it natural numbers the some office digits is 8 ok then the set of all letters of the word bit out some of all letters in the word better so will it be e T and I T or e all the repeat where Abraham said many healing here because we do not write to be their elements in a set ok clear em so I think these questions are enough I mean you see you have got an idea key the roaster fungus set builder for Magus is a set builder for host or Omega syllabus where to use this symbol belongs to does not belong store and how do we define that the collection is set or not clear on this point now there is sort of a next very important topic that is let us see different types of sex ok KRK a different data sets host there let us see I mean some very elementary definition on sets can I raise these questions okay now let us start a very important topic of sets that is different type of sense now where you see is very different type of sets that exist the first type of set that we have to see is an empty set empty or empty set going of kabhi kabhi null set yuxun void said B K there so empty set and

this is how we represent an empty set either by an empty curly braces what of file this is how way the present an empty null or a voids it now I think the name is self-explanatory to explain what this set is empty number of whites it is basically a set which does not contain any element so it s aside this liquid element exists here he cat-cow say hum empty set – null set ahead so basically it is all set with no elements a set which does not contain any element perfect with example sister a huge empty set came couch Parvati moon let us say whoever set X such that my X lies between 1 to 2 and X is a natural number okay so I say X lies between 1 to 2 and X is a national number but we know Cuban say to give each by the way national number who he named circa so this is an empty set similarly let us consider the set beam such that X X square minus 2 equal to 0 and X is a rational number from janta ki X square minus 2 Co solution the other plus minus under root 2 1 plus minus and root 2 is not a rational number which means this set cannot contain any element in it fit a core example they came X such that X is an even prime number greater gun so what is X is an even prime number greater than 2 I know this is going to be in empty set because 2 is the only even prime number that exists to say but I even prime number 2 who him he this is an empty set similarly up which real world there are examples for the path I engage my empty set see corresponding with a home and which leaves show empty set it then for example let me give you an example I say a set of all the students studying in class a set of stressed all the students studying in class 11 and 12 simultaneously as a student to 11th or 12th class my ex-husband Rahul this is not possible because we cannot have a student who is registered in class 11th and 12th in the same session true this is going to wear empty set you see that he gets to make a who I have a set of all even the wizards of seven so this is again an empty set because seven is an odd number it is not going to have any even de visite with it so this is an empty set so is where empty set kya hota hai ache sr set is my koi bhi element present now say hum ka hey empty set okay now let us see the next type of set which is finite and infinite set it’s repelling I want you to revise one definition that is number of elements in a set Homme de la casa number of element in the set is the number of distinct elements present in the said it was conquer the head number of elements in the set now a set which contains finite number of elements present in it we call it as a finite set and a set which does not contain finite number of element with infinite who say I’m in finite set K a good example take the hem let us say we have a set in X such that X is number of people present on earth okay and let us say up side B such that X is such that X is number of students in class xi a of my school up they can number of students in class 11th day of high school this is a finite set savage EQ I can count number of students that were present in a class xi a of my school obviously they would be 3048 max hundred

students so per still it is a number which can be counted so we have a finite number of elements present in the set so it is a finite set okay now consider this example number of people’s present on earth this is a finite number but a very very very large number I mean we cannot definitely say what is that number going to it can be 1 million one tweak when n number of people can be present on the earth at a time so this is what this is an in finite so basically a set with the not contained definite number of element is said to be an infinite set similarly set of natural numbers X is a natural number again this is an infinite set Karen chickie we cannot say how many elements would be present in this set we are not sure about the number of elements present and they say it is going to a very big number so we cannot say so this is an infinite set now let us say Q set of rational number again this is an infinite set on the contrary let us take a set in X such that X is a natural number less than X is a natural number less than 100 this is going to be a finite set because we can work out the number of rational numbers less than 100 that are going to be 100 so this is a definite said we are sure he is make a plain number of elements hungry so this is a finite set getting it how do we say an infinite set and a finite set couch or example Zuma poem came okay let me read a question we have you have to identify how many finite or infinite elements are there in a sec okay so the set of months of a year but they either is going to be a finite set or even find side B it is a finite set because we know there are twelve months in a year a set which is given by one five seven eleven and so on so where can I see is this is a set of prime numbers I think is my one two three we present another okay this is a set of prime numbers but this is again an infinite say because we cannot say how many prime numbers does exist okay then a set which is given by one two three ninety nine hundred so this is a set of all natural numbers less than 100 this is a finite set now the set of positive integers greater than 100 set of positive integers greater than 100 this is the infinite says because 100 sabbaticals are like keeping a positive integers whose equation the set of prime numbers less than 99 this is our finance and we can actually count how many prime numbers are less than 99 that exist so I hope now everyone is getting what is a finite set and what is an infinite set a SR said disk in number of elements from John Sacre we can say we can quote a finite number for it that is a finite set otherwise the set isn’t in finite set clear perfectly clear now let us see the next definition of equal sets when do we say two sets are equal huh number see case Mothercare is it the same key a is equal to B Jeff it too is equals to two so we can define equality in terms of numbers now let us see how do we define equality in terms of sets so topic that we’re going to see is equal six now two sets are said to be equal when they have same elements which means two sexes a and B are said to be equal if a belongs to a implies a belongs to B and

P belongs to B implies B belongs to got it those sides come come equal K page Abdul Roma exactly same elements presenter for example if I have a set a 1 2 3 4 and have a set B 4 1 2 3 then these are equal sets because they contain same elements present in them these are equal sets let us take some another example I say a set in 1 2 3 4 5 till hundred okay and I have a set in X X belongs to n X is a natural number and X less than hundred so the both the sets contain same elements so what is this these are equal elements and let us say I have a set C this is B and this C now which is X plus 1 and X belongs to 0 to 99 such that X belongs to 0 to 99 so this set Square will contain elements 1 200 so ABC are equal sets got it flare is synthetic approach real-world examples they flip ahem for example I say a is a set of two set of girls set of girls studying in class 11a okay so a is a set of girls studying in class Levin a and B is a set of girls right a set of girls from class limit a writing examination of class 11 okay so it is going to be equal sets because Jo Leduc here xi classically registered hogi xi MA my xi s examinations will EKG these are going to be same set of girls so go sets y equals public at the hem when we have equal elements and what is the definition we say set a is equals to b this is how we need to equality of two sets so we say set is equals to B if and only if a belongs to a implies B belongs to B and a belongs to B implies K belongs to our here condition satisfy Paquita be home care the hey two sets are equal this is how we define equal sets clear next let us see a very very important definition related to say that is subsets okay now little understand about the subset subset li definition becomes a pain leahy have enough for some tau what is a subset a semi limited output example scope they came let us consider two sets a and B okay a is a set of all alphabets in English okay and B is a set of all vowels in English okay this is one pair of sex next take this example a is a set of students in a school okay B here is a set of all girls in school again consider one more example here a set o all devices let us say 80 okay and be set of all

prime divisors of 80 amateur hockey okay up in the– no examples could be haunted egg here mucho bodega in the– no make a common relation gasps what is the common thing or what is a common pattern that exists between this relation this and this is set is sake or is it make in Tino’s pairs of sets with a are common hen they kill Mujibur tiger they Hagar hello SEC is example via first set of all alphabets in English set of all vowels in English a Mucha bodega ball is B alphabet hey hey hey hey Nina so AEIOU Joe is set my present Dona moça Nina me a set met Oh Artie he him so what I give keep beam as if the elements have oh ma already exist cut there him this is what we understand okay now set of all students in the school and set of all girls in the school okay do girls have Ruby student may he come through the hem to joy elements it set by present here all the girls that are already counted in the number of students in the school who’s this Sigma G three elements present ago already first while a segment exist kurta he similarly set of hold the reserves of 80 and set of all prime divisions of Italy Israel supreme lip prime divisors of 18 STS but to go be though is statement are PE who make because they are even the Wizards of eighty so be Majid the element a bow already a may present so Cal so measure hey um what we are understanding is the common relation between all these three examples is Kim’s OB element said to be may present head who already set a mayhem to make a sec you said be is contained inside a case of yogini all the vowels are contained in all the alphabets of English alphabet of English again all the girls present in a school are contained in the set all the students present in the school all the prime divisors of a they are contained as I said all the divisors of KT get it I mean this is good this is the sounding logical yapper Mecca who this B is a subset of a a car a told us a set a grammatical Oh – this is P so this is a subset of a this B set of girls is a subset of the set all the students in the school students in the school K X sub set all the girls present in the school prime divisors are the subset of the oiled divisors of number 80 quartet but some sexy inference of much power here of definition so much here the definition says E is a subset or is a subset or contained in so E is a subset or contained in B if a belongs to a implies a belongs to B is said to be a subset of B or a said to be contained and B when a belongs to a implies a belongs to B but the other coil element ma at ahead is compatible element B may go on a heating am I making sense here make head yoki coil element agar a set Miyata hey is come at la busca be set with or nahi pata cubby make aha wiki a Jo have a beaker subset head it is logical B is a big set over here and E is some small set over here okay B my elements have 1 2 3 4 5 6 what is mayhem 1 2 or 3 good Jo a V a big a subset come over there a major elements have a be me to present ho he ho tepee a bigger edge what as a subset hey cut it so this is the definition of a subset a set a is said to be a subset of or contained and set B when every element each and every element appearing in a has to be present in the set B also

denote Kasich at a be denoted by this symbol a is a subset of a is contained in P this is the symbol to represent subset okay so this open end per head the superset and close to entry hey is the subset okay got this this definition is clear now consider some situations over here Carol case at the head a is always a subset of itself a himesh of vodka subset the hoga a very clear C bar they all the elements that are present in set a are already present in set a so a is a subset of itself similarly I can say Phi is even a subset of a in fact Phi is a subset of all these X let me write it this way Phi is a subset all the sets does this make sense make it you Phi’s your hair was sublease X Kylie M sub e said come so we said subset of because Phi is nothing but empty and that has present in all the sides so Phi is a subset of all the sets up a called a key definition if a is a subset of b and b is a subset of a what this is implies this implies set a is equals to they check a is a subset of B machlup Yahoo chaga if a belongs to a implies a belongs to B and B belongs to C the subset of a cow’s leg on this embracive a belongs to B this implies B belong if a belongs to capital in so definition K away the definition is a belongs to a implies a belongs to B and a belongs to B implies a belongs to a this implies a and B are equal set UK the eco-center definition Bundy Nam which means if a is a subset of b and b is a subset of a this implies a is equals to B these two sets are equal now here is one more definition if a is a subset of B and a is not equals to B okay remember this is a subset of B but a is not equal to B then a is said to be a proper subset of P is said to be proper subset of B okay if a is a subset of B n is not equal to B then a is said to be a proper subset of b and b is said to be a super sick getting this definition if a is a subset of B and a is not equal to B then a is said to be a proper subset of b and b is said to be a superset of a but it what is the definition of a proper subset in the subset subset can be equal where a is a subset of V Kathy home so a can be equals to B but a is a proper subset which means a cannot be equal to B and then be said to be a superset of these definitions are clear subset covers of so much mile exit Mesa couch part nicolo’s oh say hum here’s a subset so egg Satan is a chode as a set nickel go to say make a tea who sub set clear him koib it out away another capita region any any small doubt pres until this point now let us consider some more examples of subsets in fact Kamaraju number system made natural numbers are integers whole numbers of eBay offers me a goose take a subset size they create guessing