 Hi and welcome to the session, I am Asha and I am going to help you with the following question which says, find the union of each of the following pairs of sets. First let us learn, what is the union of two sets? Suppose we have two sets A and B then the union is denoted by A union B which is equal to all those X such that X belongs to A or X belongs to B, that is the union of two sets A and B is the set C, A union B is equal to C which consists of all those elements which are either an A or a B including those which are in both. So this is the key idea, we are going to use in this problem to solve it, now start with the solution and the first one is X which contains element 1, 3 and 5 and Y which contains the element 1, 2 and 3, now all elements which are either an A or B or both are 1, 3 and 5, the set which contains all the elements which are either an A or B or in both which are 1, 2, 3 and 5 and that is the answer is X union Y is equal to 1, 2, 3 and 5. So this completes the first part and now proceeding on to the second part where A is equal to A, E, I, O, U, V is equal to A, V and C, all elements which are either an A or B or in both are A, B, C, E, I, A union B is equal to A, B, C, E, I, O, U. So this completes the second part and now proceeding on to the third part where A contains all those X such that X is a natural number multiple of 3, so natural number starts from 1 and the first multiple of 3 is 3, next is 6 and then we have 9, 12, 15 and so on. So this is the set A and the set B contains all those X such that X is a natural number less than 6 which is equal to the first natural number is 1 and we have to write all those natural number which is less than or equal to 6, so we have 2, 3, 4 and 5 which is a set of all those elements which are either in A or in B or in both. So these elements are 1, 2, 3, 4, 5, 6, 9, 12, 15 and so on which can also be written as X such that X is equal to 1, 2, 4, 5 or all the multiples of 3 which are natural numbers. A union B is equal to all those X such that X is equal to 1, 2, 4, 5, multiples of 3. This completes the third part and now proceeding on to the fourth part which is A such that it contains all those X such that X is a natural number, 1 is less than X is less than or equal to 6 and B contains all those X such that X is a natural number, X is less than X is less than 10. So A will be the set which contains all those natural numbers which are greater than 1 and less than or equal to 6. So they are 2, 3, 4, 5 and 6 and B will be the set containing elements all those natural numbers which are in between 6 and 10 and that are 7, 8 and 9. So A union B will be 2, 3, 4, 5, 6, 7, 8, 9. A union B will be the set of elements which are either in A or in B or in both which can also be written as all those X such that X is less than 10 and greater than 1 and also X belongs to the group of natural numbers and hence our answer is A union B is equal to X such that X is less than 10 and greater than 1 and X belongs to the group of natural numbers. So this completes the fourth part, proceeding on to the fifth part. The set having elements 1, 2 and 3 and B is an empty set. All the elements which are either in A, 2 and 3. Let having elements 1, 2 and 3, elements 1, 2 and 3. So this completes the solution. Hope you enjoyed it. Take care and bye for now.