 Hello and welcome to the session. In this session, we will learn about cardinal properties of sets. For this, first of all, consider two overlapping sets. A, this green diagram, which shows three overlapping subsets of the universal set. That is, the universal set is represented by U and it is having three overlapping subsets, which are the set A and such B. Now let number of elements in A minus B is equal to A, number of elements in B minus A is equal to B and number of elements in A into section B is equal to C. Now here, this portion of A excluding this portion, which is common to both A and B, is A minus B. Now it is given that the number of elements in A minus B is this portion of B of the intersection and number of elements in B minus A into section B. That means this region, which is common to both, is given to us as number of elements in A minus B is equal to number of elements in A minus number of elements in A into section B. That implies A is equal to number of elements in A, which further gives number of elements in A is equal to A plus C. Number of elements is equal to number of elements in B minus number of elements in A into section B. Now this implies this in A into section B is this. So this will be B is equal to number of elements in B minus C, which further gives number of elements in B is equal to B plus C. Now the next property is number of elements in A union B that means elements which are in A or in B or in A and B both equal to number of elements in A minus B plus number of elements in B minus A plus number of elements which are in this common portion. That means number of elements in A intersection B. So this is equal to number of elements in A minus B plus number of elements in B minus A plus number of elements in A intersection B. Now other properties which we have discussed earlier elements in A union B is equal to number of elements in A minus V is equal to number of element minus number of elements in A intersection B plus number of elements in B minus minus number of elements in A intersection B plus number of elements in A intersection B. Now, these are cancelled in each other. So, this implies number of elements is equal to number of elements in A plus number of elements in A intersection from this ring right is equal to number of elements A complement which are in the universal set but not in A. Therefore, is equal to number of elements in A plus number of elements in A complement. Number of elements in the universal set is equal to number of elements in B plus number of elements in B complement. B complement is a set of those elements of the universal set which are not in B. So, this will be equal to number of elements in B plus number of elements in B complement. Now, let us discuss an example by using the cardinal properties of set hundred people in a restaurant where 50 people like continental food, 25 like darling food and 30 people do not like any of these kinds of only continental and therefore how many like. Now, let us start with its solution by using cardinal properties of sets that in a group of hundred people in a restaurant 50 people like continental food, 25 like Italian food and 30 people do not like any of these kinds of foods. Now, let set is represented by U which is equal to the set containing people. Number of elements in A is equal to and let B is equal to the set containing people. Now, the people who like Italian food are 25 therefore number of elements in B is equal to 25. U in diagram is equal to 50. Number of elements in B is equal to 25. People do not like any of these kinds of food. That means those elements will be inside this rectangle. Now, number of elements in A union B is equal to number of elements in universal set which are hundred minus number of elements which are outside this portion of A union B rectangle which is representing the universal set. So, this is equal to which we have discussed earlier in A union B is equal to number of elements in A plus number of elements in B section B is equal to this in A plus number of elements in B minus number of elements in A union B. So, here number of elements in B are minus 17 which is equal to 5 elements which are common to both A and B are number of people in A minus B is equal to number of elements in A minus number of elements in A intersection B and number of elements in A intersection B is equal to 5. So, putting these values here it will be 50 minus 5 which is equal to 45. Therefore, the number of people that means these are the elements of B which are not in A that is these are the people who like Italian food only. Now, B minus A is a set containing the people property. Number of elements in B minus A is equal to number of elements in B minus number of elements in A intersection B which are equal to now number of elements in B is equal to 25 values here it will be 25 minus 5 which is equal to 20. Therefore, properties of sets enjoy the session.