 Hello everyone, I am Rachna Pathak from Walshan Institute of Technology, Singapore and today we will study most important topic according to gate point of view. So these video series will help all the aspirants preparing for gate to 2020. It's important to focus on the script mathematical structure as every year statistics says DMS wakes for 14 to 16 marks and the average weightage is 8 to 12 person. So now we can imagine how important it is to study discrete mathematical structure. In this video, we will study all about relations in discrete mathematics. Learning outcome. At the end of this session, student will be able to explain all about relation, domain and range in a relation and how to solve the problem based on relations. Now let us see what are the prerequisites. Basically you should know all about basic structures and discrete mathematics basic. Student from CAC or IT Engineering branch gate aspirants those who are preparing for CS paper. Now beginning with the subject. So the idea of relation is a basic concept that is widely used in mathematics as well as in our day to day life. We all are familiar and let us see a simple example of what a relation is. Now what is a relation? We see the relation can be a father to a son. It can be from a father to a daughter. The relation can be from a mother to a son or it can be from a mother to a daughter. So these are some types of relation. Now let us describe in terms of discrete mathematical structure. Consider a set A and let us consider another set B. In set A let the element be father and mother. In set B let the element be son and daughter. So here we can see father can be related to a son. Father can also relate to a daughter. Mother can relate to a son and mother can be related to a daughter. So now this can be further ordered as we see the element from one set is ordered or forms a pair with another set. So we can see father son, father daughter, mother son and mother daughter. Now this is set ordered pair as you cannot swap or change the order. Example you cannot relate son to a father. So this inversely is not possible. Let us see more in detail. Now we can say the collection of ordered pair symbolizes association from members of one set to the members of another set. And we can say the set that contains these ordered pair are also called as binary relation. Now I have a question for you. Can we show the relation between son and daughter? As per the previous example, take a pause so the answer is yes because even they are element of the same set you can make a copy of a set and then pair them to form a particular set. Of course in some cases the meaning of the relation needs to be modified. You can also say that these are similar to Cartesian product where you see all the possibilities from set A to set B and these are presented or represented as A Cartesian B that is nothing but I can show as A into B. So now coming to traditional definition of relation is any set of ordered pair defines a binary relation or simply a relation. Binary relations represent relationship between element of two set. Now if you consider example of relation a particular ordered pair x, y belongs to a set A and it belongs to a particular relation. This can be represented as x capital R y and can be read as x is in relation to R with y. So this is how we traditionally define relation. Now let us consider a simple example. The relation greater than for real number is denoted by the symbol greater. Now if x and y are any two real number such that x is greater than y then we say that x, y belongs to the relation greater than. So this is how we represent mathematically or in terms of symbolic format. Now does the relation greater is greater than that is your relation equals to x, y such that x and y are real number and your x must be greater than y. So now this symbol is used to represent such that this method of representation is called a set builder method. So basically there are two types of method one is roster tabular method and another is your set builder method. So we are not going to see much in detail about it but just for your information this is your set builder method. Now let's move to one more concept of domain and range of a relation. Domain of all the first number set of all the first number of the order pair is nothing but your domain. In other words the domain is all of the x values. Now let's see what is range. Set of the second number in each pair or in simple term all the y values. Let us see the example where we identify the domain and the range. Example domain and range of relation. Now let us see I have a set 0, 1, 3, 22, 90, 34. So this is a set. Now what is the domain? Domain are all of your x value like 0, 3 and 90. This will fall into domain and 1, 22, 34 are your range that is your y values. So this is the basic concept of your domain and range. Let us see one more example. List the domain and the range of the relation. So your question is minus 3, 3, 0, 2, 3, 3, 6, 4 and 7, 7. So the domain will be all the x values minus 3, 0, 3, 6 and 7. Range will be 3, 2, 3, 4 and 7. Now one thing which is to be considered or observed over here if you have two same values the range will consider only one value and this is possible for both the cases. Suppose I have minus 3, minus 3, minus 3 but when I list down my domain values it will be appearing only once that is minus 3. Only the one time minus 3 will occur over here. Even though the number 3 is listed twice right in the relation you only noted once that the domain or range it can be either your domain or range we just write down once. Now representation for a set S. So this is how your set is represented. The domain of S equals to x such that there exists y where x, y belongs to S. So this is for domain of S. In mathematical terms your range is represented as range of S that is r of S equals to y such that there exists x where your x, y belongs to S. So this is how we represent domain and range, conclusion. Now in this session we have studied all about the relation. We have studied about domain ranges and what your relation is how you can represent in mathematical terms. So here are some of the references which I have used during creation of this video. Thank you.