 చాకారాటణ ళనికామని Author Roller ని� þ�రూడినిలాటి. screws కాకికిటాబ లిమాకండి కాకిలూతమనికాక రికి� immediately మానివరియా లికustomed ఆతికారార మానికింటినం. మాన౿టికింటి, ఇత౿కికిరినికి� を న౿రి ఆఁరిటికికికికాట So first we understood some, we discussed some example of well defined collection, the collection of all course of art at bazookan toぇ hence state or open university, ఱాడిగారక కరక。」 ని స఩టిల crise నిగా� hanging సండంచతటారం�短రభు ప౒తూచాతోటా �繼續టినసంచంచారుగి కిఎస్రయాధరిు ఱఆర్రాక Dansh మారిధరలారిండియాతోచం. ఇరివాన్దితమా సకానితానిదిండిదా నిందాడిందాని. ఠినానూడిఉాని. నితినిక౒లానిని. మూదికనుసంలండిది. మరిని తిని సిసినిండి. అధిలారి లిక౿స్టక కిసిలి. నాతిములి మ్ంపిని.. సిలోస్టకిలినానేయాది. ఴికశానేముకిటీనికి. So we cannot say that these collections are examples of set, so these collections are not examples of set. So for a set it should be a well defined collection of objects. So next we define set memberships. The objects in a set are called as member or elements. Sessions are generally denoted by captain letters. So we use captain letters to denote a set. This captain letter, capital A, B, C, D, X, Y, Z. This captain letter we can use for representation of a set. Elements of a set. The elements of a set are denoted by small letters. A, B, C, X, Y, Z, etc. The set memberships express, set memberships is expressed by using Greek letter epsilon. This symbol is epsilon. So suppose X is an element of a set A. So then we write X epsilon A. This means X belongs to A. And if X is not an element of A, we can say that X does not belong to A. We can use this symbol. Here one example for set memberships. Suppose we consider a set A. Here elements are 1, 3, 5, 7, 9. Then share 5 belongs to this set but 6 does not belong to this set. So we can denote this symbol epsilon. So 5 belongs to A but 6 does not belong to A. Then we discuss how to represent a set, representation of a set. So there are two methods for representing a set. The first method is set tableaux method. In this method we list all the members of the set separating them by commas and closing them by is in curly breakage. So this is the representation of a set in tableaux form. In tableaux form we list all the members of the set and separating the elements by a comma and enclosing them in curly breakage. The examples, here are some examples. How can we represent a set in tableaux method? If A is a set of prime number less than 15, here A is the collection of all prime number less than 15. Then we can represent this set in tableaux form this way A. So all prime number less than 15. So we start from 2, 2, 3, 5, 7, 11, 13. So we represent this set in tableaux method. Another example, if he is the set of howling in English alphabet so we write all the vowels and we list all the vowels of the set and separating them by commas and enclosing them by them in curly breakage. So we get this set in tableaux form method A, E, I, O, U. So this is the set of howling in English alphabet. So next representation of a set is set builder method. So in this method we write the set by some special property and write it as this way. So in set builder method we do not list the elements. We just represent, we put some special property. Here you see A, X is the element of A such that X satisfies this property P of X. P of X is some property or maybe some formula maybe this. So X such that X is the property P X. So X represent all the elements and X has some property. And if it has A is the set of all elements X such that X is the property P. Here some examples of set which are represented in set builder method. Suppose here set is given in tableaux method 1, 2, 3, 4, 5, 6. So we can write this in set builder method this way. X, so X is the element of A such that X belongs to N. N means set of natural number such that X less than 7. Here 1 to 6 elements from 1 to 6 and which are natural number. So we write this X satisfies this property X less than 7 and X is a natural number. And another example is the set of all event numbers lying between 1 and 31. 1 between 1 and 31. So set of all event numbers. So all event numbers will be between 1 to 31 and we can write this set in set builder method this way. A X, X is the element of A such that X satisfies this property. 1 less than X less than 31 because X is the event number between 1 and 31 and we write X is the event. So next we discuss some types of set types of set the first null or empty or void set. A set having no elements is called a null set means a set where there is no element. And it is denoted by the notation this crawler bracket and here space is blank or phi where this symbol is a Greek letter phi. So A set having no elements is called a null set and it is denoted by a Greek letter phi. Example the set of human beings living in moon. So this set has no element because there is no human being living in moon. So this is a null set X such that X belongs to n and 4 X lies between 4 and 5. So there is no element between 4 and 5 which is natural number. So this is a null set. The set of all prime number divisible by 2, 3, 5. So there is no prime number which are divisible by 2, 3 and 5. The set of months in a calendar is starting with the letter p. So there is a no month in a calendar which is start with the letter p. So this is the set of null set. So next type is finite and infinitive. A set is finite if it contains finite numbers of defined element. Means the elements can be, we can count the number of elements. Then that set is said to be finite. A set which is not finite is called an infinite. So example of finite set, the set of holes in English alphabet, there is a 5 letters, 5 holes. So therefore these are finite set. The set of rivers in India is also finite set. An example of infinite set, the set of all points lying in a straight line. So in a straight line there are infinite number of points. The set of all is thirst in the sky. So in the sky there are infinite set. And n, the set of missile number 1, 2, 3, 4, 5 after infinity is also an infinite set. So next type of set is subset. If every element of A, suppose we consider a set A. In every element of set A is also an element of set B, then A is said to be, is called a subset. And we write this as this symbol subset, A is subset of B, which is written as A is subset of B. Or A is contained in B. So A subset of B, this implies that implied by this curly bracket, X belongs to A implies X belongs to B. When X belongs to A implies X belongs to B, then we can say that A is a subset of B. Then there is two types of subset, one is proper subset and another is improper subset. The proper subset, if A is a subset of B and A not equal to B, then A is a proper subset of A. And we write this as this symbol. This symbol means proper subset. The equality sign is not there. So it is a proper subset. Then improper subset, the null set phi, the null set already we define what is the null set. The null set phi is a subset of every set. And every set is a subset of itself. That is phi is a subset of A. And A is a subset of A itself or every set A. So they are called improper subsets of A. Because every null entry set has two improper subsets. So naturally define universal set. If set has all its subset and their elements, then the set is called an universal set. Universal set is generally denoted by capital letter U. So we take one example for universal set. Let P is a set of all prime numbers. O set of all odd numbers. E set of all even numbers. And N set of all natural numbers. So here there are four sets. And here larger set is N. So we can say that N is the universal set. Because P, O, U, all the sets contain N. So next we define power set. The power set means the set of all subset of a given set A is said to be the power set. And it is generally denoted by P of A. So power set means set of all subset of a given set. And it is denoted by P of A. P means power and A is set, given set. Here one formula how to find the number of elements of power set for a given set. If A set A has M elements that is N of A here N means number of elements of A equal to M. Then number of elements of power set P of A is equal to 2 square M. So this is the one important formula to find the number of elements of a power set for a given set. Here we take some examples how can we find power set of a given set. Suppose here A has elements A. A has one element, only one element. So therefore number of elements of power set is 2 to the power 1 according to this formula. 2 to the power 1 so is equal to 2. And power set P of A is equal to 5. One element will be 5 because null set is a subset of every set and that set itself. If another example number 2 let A has two elements A and B. So number of elements of P of A is 2 to the power 2 is equal to 4. And set power set is is equal to 5, element A, element B and element AB. So there are 4 elements in power set of A. So here some reference of this unit. So you can follow these books for further readings and for better understanding of this concept of set. So now we discuss some questions. Discussions of some questions. So questions 1 determine which of the following descriptions represent a set. In case the description represent a set, write a set in table or set builder. There are 2 collections are there. One collection is collection of all national number that are greater than 2. And collection of good musician of our country. So in A here given description represent a set because collection of national number that are greater than 2. This collection is well defined. And the set A we can write this in table or from this way. A is equal to 3 greater than 2 so 2. So 3 onwards 3 4 5 6 7 up to infinity. So it is also an infinite set. And collection of good musician of our country. This collection is not well defined. Because this collections person to person varies this collection. Because for me good musicians may be different collection from your collections. So it will vary from person to person. So this is not well defined collection so this collection is not a set. Next question to A here set A have 3 elements ABC. So you have to find how many subsets power set of power set A can have. Here you see number of elements of A is 3 so power set according to this formula. If a number of elements of A has m elements then number of power set of A is 2 to the power m. That formula we follow then P of A is equal to 2 cubes so 8 elements is there. And power set of this set if we take power set of this set then we take P of A then it will be 2 to the power 8 so it will be 256. So next we take another question 3. So that set A set A has 4 elements 2 3 5 7 is not subset of B. B is a set such that X belongs to N and X is odd. So here you see 2 belongs to A elements 2 belongs to A but 2 does not belong to B because B consists of all odd numbers. So 2 is an even number but B consists of odd numbers so therefore A is not a subset of B. So next question number 4. So that set A so set A has 4 elements 3 5 7 8 is power subset of B. B has this elements 1 2 3 4 5 6 7 8 9. So we have to show that A is a power subset of B. Power subset means A contain in B but A not equal to B. So here you see each element of A is also an element of B. 3 5 7 8 is also belongs to B. So each element of A is an element of B so A is a subset of B. But you see the elements 1 2 4 6 9 belongs to B but these elements 1 2 4 6 9 does not belongs to A. So hence A not equal to B. So therefore A is a power subset of B. So we sum up today's class. So here first we define a set. So set is a collection of some well-defined objects. Capital letter of English alphabet are usually used to denote the set. The small letter of English alphabet are usually used to denote the members in a set. Then we define null set. A set which does not contain any elements is called a null set. And null set it is denoted by regulator 5. And set which contain a different number of elements is called a finite set. Otherwise it is called an infinite set. So next we define a subset. If every element of a set is also an element of set B. Then we set it A as a subset of B. And we write this symbol A. This symbol in subset A is a subset of B. Which is read as A is a subset of B. And then we define power set. The set of all subset of given set is called a power set. So now we come to end of today's class. So in next class we will meet again with unit 15. Thank you all.