 Hello and welcome to the session on the topic set theory under the course discrete mathematical structures at second year of information technology engineering semester one. In this lecture, students will be able to apply principles of set theory to solve problems involving sets. The flow of the session will solve certain examples first and we will discuss the solutions and you will be given an assignment to solve on your own. So, here is the first example. Here we are solving certain examples based on an important set operation called as Cartesian product and this is useful in many of the applications of computer. You will learn the same how it is applicable in databases to find out the required data. So, here let us solve certain examples based on Cartesian product operation. The first example, if A is equal to alpha, beta, B is equal to 1, 2, 3, find A cross B, B cross A, A cross A, B cross B and A cross B intersection B cross A. So, here is the solution. We will use the Cartesian product operation to solve each one of these. See the order. First we need to find out A cross B and we know that when you apply Cartesian product operation it simply combines each and every element from the first set with each and every element from the second set resulting into a set of ordered pairs where first element is from the first set and second element is from the second set. So, we will solve A cross B first. We observe A contains two elements alpha and beta, B contains three elements 1, 2 and 3. So, we will start writing the pairs where we get first element. See the first element from A is alpha and we will combine that with each and every element from set B. So, we get alpha, 1, alpha, 2 and alpha, 3. Similarly, now the second element from set A is beta. We will combine that with each and every element from set B resulting into beta, 1, beta, 2 and beta, 3. So, this is the answer for the first part A cross B. Similarly, we will find out B cross A. See the order now. We have to take the first element in the ordered pair from set B and the second element from set A. So, we will start with the first element of B that is 1 and we get 1, alpha and similarly we combine with the remaining element as 1, beta. Next we combine the element 2 from set B with each of alpha and beta resulting into 2, alpha and 2, beta. Similarly, the third element 3 from B is combined with alpha and beta giving us 3, alpha and 3, beta. So, this is the answer for B cross A. Now what next we have been asked is A cross A. This is the Cartesian product of the given set with itself. So, in both the places in a ordered pair, we will assume the elements coming from the same set. So, we will take alpha and beta and combine that with alpha and beta itself. So, resulting into alpha, alpha, alpha, beta, beta, alpha and beta, beta. So, this is the answer for A cross A. Now the next part B cross B is equal to B get. Now we have to assume set B and combine with itself that is 1, 2, 3 combined with 1, 2, 3 resulting into 1, 1, 1, 2, 1, 3, 2, 1, 2, 2, 2, 3, 3, 1, 3, 2 and 3, 3. So, these are the sets and finally what we have been asked is the already found set A cross B take intersection with B cross A. So, that results into we have found A cross B and we have found B cross A and then if you try to take intersection obviously we find there are no common elements. So, it results into phi that is the empty set. So, that is why we derive an important property of Cartesian product saying that A cross B is never equal to B cross A that is Cartesian product is never commutative in nature. So, with this we go on to the next example. Here is the second example. If A is a set given as phi that is an empty set and B has been given as 1, 2, 3 then find A cross B and B cross A. Simple enough we have been given one set out of the two which takes part into a Cartesian product operation as empty that is phi and then if you take A cross B as well as B cross A one of the set contains no element because in A cross B A contains no element in B cross A A contains no element. So, both these Cartesian product will result into phi being the empty set. So, here you have to simply assume that no elements exist in one of the sets you cannot combine with anything coming from the other set. So, that is why the answer for both A cross B equal to B cross A is equal to phi the empty set. See the assignment question for all of you given A equal to set of 1 B is equal to set of A comma B and C is equal to 2 comma 3 you have to find out all these Cartesian products A cross B cross C B square. So, how will you find B square? It is simply B cross B similarly A cube what will be A cube it is A cross A cross A then B square cross A that means you have to assume it to be B cross B first and then cross A and finally A cross B. So, these are the various Cartesian products you have to apply on the given sets A, B, C. Thank you.