 Hello and welcome to the session. In this session first we are going to discuss about Cartesian products of sets. First let's see what is an ordered pair. A pair of elements grouped together in a particular order is called an ordered pair. Like this is an ordered pair in which the elements are A and B. Now let's define the Cartesian product of two sets given P2 to be any two non-empty sets where Cartesian product is given by P cross Q and this is the set of all ordered pairs of elements from P and Q. That is this can be written as the set PQ such that P belongs to the set P and Q belongs to the set Q. So this is the Cartesian product of two sets. If we have either P or Q is a null set then the Cartesian product of P and Q that is P cross Q will also be empty set that is equal to 5. We have some important remarks like two ordered pairs like AB and CD are equal if and only if the corresponding first elements are equal that is A is equal to C and the second elements are also equal that is B is equal to D. The next is if we have that the number of elements in a set A is given to be P and number of elements in a set B is Q then number of elements in A cross B that is in the Cartesian product of the sets A and B is P multiplied by Q. And if we have A and B are non-empty sets either A or B is an infinite set then so is A cross B. Next is A cross A cross A is equal to the set A, B, C such that A, B and C are the elements of the set A. And here this A, B, C is called an ordered triplet. Consider set A equal to 1, 3, 5 and set B equal to 2, 3 then A cross B is equal to 1, 2, 1, 3, 3, 2, 3, 3, 5, 2 and 5, 3. And B cross A would be equal to 2, 1, 2, 3, 2, 5, 3, 1, 3, 3 and 3, 5. Now from these two we have that A cross B is not equal to B cross A since 1, 2 is not equal to 2, 1 and etc. Next we shall discuss relations. Our relations are from a non-empty set A to a non-empty set B is a subset of the Cartesian product A cross B. The subset is derived by describing the relationship between the first element and the second element of the ordered pairs in A cross B. The second element in the ordered pairs of the Cartesian product A cross B is called the image of the first element. Then the set of all first elements of the ordered pairs in a relation R from set A to set B is called the domain of relation R. And the set of all second elements in a relation R from a set A to a set B is called the range of the relation R. And the whole set B is the co-domain of the relation R and we have that range is a subset of co-domain. A relation may be represented by roaster method or seg welder method. By this method we can represent relation algebraically and then it can also be represented by an arrow diagram which is a visual representation of a relation. The total number of relations that can be defined from a set A to a set B is the number of possible subsets of A cross B. And if we have number of elements in set A is B and number of elements in set B is Q then number of elements in A cross B is PQ and total number of relations is equal to 2 to the power PQ. Consider a set A equal to 1, 2, 3, 4, 5, 6. Now we define the relation R on A by R equal to XY such that Y is equal to X plus 1. So the relation R in the roaster form is given by 1, 2, 2, 3, 3, 4, 4, 5, 6. We got this using this relation where we put different values for the X which are the elements of the set A and thus getting the corresponding value for Y. And this is the representation of the relation R using the arrow diagram where this is the set A which has elements 1, 2, 3, 4, 5, 6. Now since we have defined the relation R on A itself, so the set B also has the same elements as the set A. Now the domain of relation R is equal to 1, 2, 3, 4, 5. Since we know that domain is the set of all first elements of the related pairs in a relation, then co-domain of relation R is equal to the set 1, 2, 3, 4, 5, 6. Since we know that the whole set B is the co-domain of a relation R but in this case the set B is the set A itself so this is the co-domain of R. Then we have range of relation R is equal to 2, 3, 4, 5, 6. Since range is the set of all second elements in a relation R. This completes this session. Hope you have understood the concept of Cartesian product of sets and relations.