 Hello and welcome to the session. In this session, we will discuss the following question and the question says, let P is equal to the set containing the elements 2, 4, 6, 8, 10 and Q is equal to the set containing the elements 1, 3, 5, 7, 9, 11. Let R be a relation is one less than from P to Q. Find R, show it by an arrow diagram and also by an equation. Now before we start solving the question, let us first recall what is a relation? A relation is the association that exists between the first and second components of a set of ordered pairs. Let us now recall what is a domain. Domain is the set of all first components of all the ordered pairs of the relation. Now let us review what is range? Range is the set of all second components of all the ordered pairs of a relation. So this is the key idea for this question and using this key idea, we shall solve the question. Let's start the solution now. We are given that R is a relation is one less than from P to Q that is first components are elements of the set P and second components are elements of the set Q and first component is one less than the second component or we can say that in the relation R, second component is formed by adding one to the first component. So we can say that R is equal to the set containing ordered pairs x, y such that y is equal to x plus 1 where x belongs to P and y belongs to Q. So we have to form those ordered pairs x, y in which x is one less than y. We are given P is equal to the set containing the elements 2, 4, 6, 8, 10 and Q is equal to the set containing the elements 1, 3, 5, 7, 9, 11. So we form ordered pairs from the set P and Q in which first components are elements of the set P and second components are elements of the set Q and first component is one less than the second component. Therefore, R is equal to the set containing the ordered pair 2, 3, ordered pair 4, 5, ordered pair 6, 7, ordered pair 8, 9 and ordered pair 10, 11. So the relation R is equal to the set containing all these ordered pairs in which the first component is one less than the second component. Now using the key idea domain is the set of all first components of all the ordered pairs of the relation. So the domain of R is equal to the set containing all the first components of these ordered pairs. So the domain of R is equal to the set containing the elements 2, 4, 6, 8, 10. Now going back to the key idea again range is the set of all second components of all the ordered pairs of a relation. So the range of R is the set of all second components of all the ordered pairs of the set R that is range of R is equal to the set containing the elements 3, 5, 7, 9, 11. We will now draw the arrow diagram to make the arrow diagram for the relation R. We first draw two ovals in which the first oval represents the domain of the relation R and the second oval represents range of the relation R. We now write all the elements of the domain in the first oval. So we write the elements 2, 4, 6, 8, 10 in the first oval and we write the elements of the range in the second oval that is we write the elements 3, 5, 7, 9, 11 in the second oval. Next we draw these arrows linking the elements in the domain to the elements in the range. Each element in the domain is linked to its corresponding element in the range by the help of these arrows. With this we end our session. Hope you enjoyed the session.