 Hi and welcome to the session, I am Asha and I am going to help you with the following question which says if A is a set having elements 1, 2, 3 and so on up to 14 define a relation r from A to A where r is equal to order pair x, y such that 3x minus y is equal to 0 x and y are elements of A, write down its domain, co-domain and range. So first let us learn some simple definitions with the help of which we will solve the above problem. Let us start with the term relation. Suppose we have any two non-empty sets A and B a relation from A to B is a subset of the Cartesian product A cross B. Now the subset is derived by describing a relationship between the first element and the second element of the ordered pair A cross B. The second element is called the image of the first element. The relation r which is the subset of the Cartesian product A cross B and is derived by describing a relationship has ordered pairs x plus y such that x belong to A and y belong to B. And now let us come to the term domain which implies the set of all the first elements of the ordered pair. In a relation r from a set A to a set B is called the domain and range is the set of all the second element of the ordered pair. Co-domain the whole set B and range is always a proper subset of the co-domain. So these are some definitions which we should always keep in mind before solving problems of this type. This is a key idea. Let us now begin with the solution and here we are given a set A with elements of 1, 2, 3, 4 and so on up to 14. And we are required to define a relation from A to A. The relation r is defined by all the ordered pair x comma y such that the relation between x and y is 3x minus y is equal to 0 where x and y belong to A. This is the subset of the Cartesian product A cross A to define the ordered pair which satisfies the condition 3x minus y is equal to 0. Now, first let x is equal to 1 then 3 into 1 minus y is equal to 0 implies y is equal to 3. Similarly, when x is equal to 2, 3 and 4 this implies y is equal to 6, 9 and 12 since our relation y is equal to 3x. Now, if we take x is equal to 5 then y will be equal to 3 into 5 which is equal to 15 which do not belong to the set A. Hence, x comma y if equals to 5 comma 15 this does not belong to the relation r. Thus the relation r will have elements 1, 3 that is when x is 1 then y is 3. When x is 2 y is 6, when x is 3 y is 9 and when x is 4 y is 12. Now, here is find the domain of r. Domain of r will be the set of elements which are the first elements of these ordered pairs. First elements are 1, 2, 3 and 4. Let us find the range of r. Range of r will be the set of elements these ordered pairs which are the second element as 3, 6, 9 and 12 and now lastly we have to find the co-domain of r which will be the set A itself that is the set having elements 1, 2, 3, 4 up to 14. So, this is the answer. This completes the solution. Hope you enjoyed it. Take care and bye for now.