 Hello and welcome to the session. In this session first we will discuss about sets. The concept of set serves as a fundamental part of present-day mathematics. It is used to define the concepts of relations and functions. Basically a set is a well-defined collection of objects and the sets are usually denoted by capital letters like A, B, C, X, Y and so on. And the objects of the set or you can say elements of the set are represented by small letters like A, B, C, X, Y, Z and so on. We know that N is the set of natural numbers. This is an example of the set. If we have a set A and an element A, if we want to say that this element A is of the set A, we write it in this form. This means that the element A belongs to the set A and if we have an element B and a set A and this element B does not belong to set A we write it in this form. That is B does not belong to the set A. Then next we have two methods of representing a set. The first one being roaster or tabular form. In this form all the elements of a set are listed. The elements are separated by commas and are enclosed within races. Consider a set A which is the set of all odd natural numbers less than 10. We know that odd natural numbers less than 10 are given by 1, 3, 5, 7, 9. So in the roaster or tabular form the set A is written in this form. That is all the odd natural numbers less than 10 are written within the braces and they are separated by commas. And in the roaster form the order in which the elements are listed is immaterial. That is this set can also be written as 1, 5, 9, 3, 7. Order does not matter in this case. The next method to represent a set is set builder form. In this form all the elements of a set possess a single common property which is not possessed by any element outside the set. This is written in the form X that is the element of the set which is X. Then we put a colon and after the sign of colon we write the characteristic property possessed by the elements of the set and then we enclose the whole description within races. Or we can also say X satisfies properties P and this is written as the set of all those X such that each X satisfies the properties P. Like suppose we consider a set A equal to 1, 2, 3, 4, 5, 6, 7. As you can see that this set A is the set of all natural numbers less than 8. So in the set builder form this can be written as X colon that is we can also say X such that X belongs to N that is X is a natural number and X is less than 8. This is the set builder form of the given set A. Next we have empty set a set which does not contain any element is called the empty set or the null set or the void set. An empty set is denoted by the symbol phi or this. Consider a set A equal to X such that X belongs to N and X is greater than 2 and less than 3. Here we have that X is a natural number which lies between 2 and 3 but we know that there is no natural number between 2 and 3 therefore we can say that A is an empty set. Next is finite and infinite sets now a set which is empty or consists of a definite number of elements is called a finite set and then a set which is not finite is called infinite set. If you consider a set A equal to 1, 2, 3, 4 now as you can see that this set A consists of finite number of elements that is 4 so we can say that the set A is a finite set. Now N is the set of all natural numbers that is this set. Now this set N is not finite since it does not have definite number of elements so we say that the set N is infinite set. Next we have equal sets 2 sets A and B are equal if they have exactly same elements and we write it as A equal to B and if the 2 sets are not equal they are called unequal sets. And this is written as A not equal to B. Suppose that we have a set A equal to P, Q, RS and set B equal to Q, R, P, S. Now as you can see that the elements in both these sets are same but just their order is different so we can say that A is equal to B that is the set A is equal to set B. Next we have subsets. A set A is set to be a subset of set B if every element of A is also an element of B. This is the symbol for the subset so when we say that A is a subset of set B we write it in this form and if we have that an element A belongs to a set A then A belongs to the set B whenever we have A is a subset of B. A is not a subset of B is written in this form and we also have that every set is a subset of itself that is a set A is a subset of itself. And we also have that empty set phi is a subset of every set and a set which is only one element like this is called a singleton set. Consider a set A with elements 2, 3, 5 and a set B with elements 2, 3, 5, 7, 9. Now as you can see that every element of set A is an element of set B so we can say that A is a subset of B. Next we have subsets of set of real numbers. We have many important subsets of real numbers that is R like N which is the set of natural numbers is a subset of real numbers R. Then set of integers Z is a subset of the set of real numbers R. Then we have set of rational numbers Q is a subset of real numbers R and set of irrational numbers T is a subset of real number R. Then we have a very important relation which says that N which is the set of natural numbers is a subset of set of integers Z which is further a subset of set of rational numbers Q and also set of natural numbers N is not a subset of set of irrational numbers T. Then we have intervals as subsets of R. Consider A and B belong to R and we have A is less than B. Then the open interval A B denoted by this is the set of real numbers Y such that Y is greater than A and less than B. As you can see all the points between A and B belong to the open interval A B but the points A and B then says do not belong to this interval. Now the interval which contains the end points also is the closed interval which is denoted by this that is closed interval A B. This is equal to the set X such that X is greater than equal to A and less than equal to B. That is this interval contains all the points between A and B and also the points A and B themselves. Now we also have intervals which are closed at one end and open at the other end like this interval which is open at B and closed at A. This is equal to the set X such that X is greater than equal to A and less than B. This is an open interval from A to B including A but excluding B. Also we have another interval of this kind which is equal to the set X such that X is greater than A less than equal to B. This is an open interval from A to B including B but excluding A. And the number B minus A is the length of any of the above four intervals. Like the closed interval 2 3 is the set X belongs to R such that X is greater than equal to 2 and less than equal to 3. Next we have power set. The collection of all subsets of a set A is called the power set of A. It is denoted by P A. In P A every element is a set. Let's consider a set A which has elements 2 3. Now the subsets of set A are given by phi, single term 2, single term 3 and set 2 3. So the power set of A given by P A is equal to phi, 2, 3 and 2 3. So this is the power set of the given set A. And also if the number of elements in a set A is equal to M then the number of elements in the power set of A is equal to 2 raised to the power M. Now like in this set we have M equal to 2 that is we have number of elements in this set as 2. So the number of elements in the power set of the given set would be 2 raised to the power 2 which is equal to 4. And as you can see we have 4 elements in the power set. Next we discuss universal set. Set of all elements under consideration is called universal set. It is usually denoted by the capital letter U and its subsets are denoted by the letters A, B, C etc. Like for the set of all integers the universal set U can be the set of rational numbers and for that matter set R of real numbers. This completes the session. Hope you have understood the concept of sets.