 Hello everyone, I am Richna Pathak from Walshan Institute of Technology, Sholapur. So today we are going to see in this session all about relations post-it. Now let's begin with what is post-it? Learning outcome. At the end of this session, student will be able to explain partial order relation that is nothing but abbreviated as post-it in relations. Now what are the prerequisites for this? First of all you should be aware about the basic discrete mathematical structure, basics of relation which you can see in video one if you want to refer. Now beginning with the introduction. So what is a post-it? Post-it is nothing but partial order relation. Set which is partially ordered. A partial ordered set is nothing but a post-it. Now let us take a simple example for it. Consider a relation R on a set A which is called as a partial order if it is reflexive, anti-symmetric and transitive. Yes, here I mean to say you should satisfy all the three conditions that is reflexive, anti-symmetric and transitive. Now many students get confused with equivalence of relation and post-it. Post-it satisfies these three conditions, reflexive, anti-symmetric and transitive. The set A together with the partial order R is called a partially ordered set or simply a post-it which you can denote by A comma R. Now let A be a collection of a subset of a set S. The relation subset of set inclusion is a partial order set or partial order on A so that you can write as A comma subset is a post-it. So this is how we define all about post-it. You will get a more clear concept when I use examples to explain. So basically we need to remember there are three important properties which is must to satisfy a particular set as a post-it. And those properties are nothing but reflexive, anti-symmetric and transitive. So if you satisfy these three properties then a set is said to be a post-it. Or I can say when a relation satisfies all these three properties then it is said to be post-it. So this is partial ordering relation. Let us take an example. Now consider a set A equals to 1, 2, 3. Now when can I say that a particular relation R1, R2, R3 and many more. Now I will identify whether this relation R post-it or not. So let us take with an example. Now suppose I have a set A where the elements in set A are 1, 2 and 3. Now I have to prove that a relation R1, initially I will take an empty set. A relation R2 where I will take elements 1 comma 1, 2 comma 2, element 3 comma 3. So I will complete this. So these are nothing but the relations R1, R2 which is formed by the set A. Now let us see or let us check whether these relations are post-it or not. So how to begin? Now I will give more clear example. Let us consider. Now the set is 1, 2, 3. It means there are 3 elements in a set A. So I can represent this as 1, 1, 1, 2, 1, 3. 2, 1, 2, 2, 2, 3 and one more it will be 3, 1, 3, 2 and 3, 3. So this is nothing but my matrix. Now let us see how you can form or how you can satisfy the conditions. Now suppose if I have reflexive property. Now when do you see that a particular relation is reflexive? When you have all the diagonal elements in a particular relation? Means if it is 1, 1, 2, 2, 3, 3. It means your i and jth values are same. Then this is said to be reflexive. So when I consider R1 I have nothing to do because it is an empty set. So you cannot do anything for an empty set. And empty set is neither reflexive nor transitive nor anti-symmetric. So here I cannot satisfy any of the condition and this is not a post-it. Now let us consider the second example. For second example we have 1, 1, 2, 2, 3, 3. So these are nothing but the ordered pairs, right. So in this relation R2 I have 1, 1, 2, 2, 3, 3. Now let us check 1, 1, 2, 2, 3, 3. Yes, this is reflexive, yeah, this one is reflexive. Now another property is your anti-symmetric. We have one more property which you have to satisfy to check whether a set is post-it or not. So we have anti-symmetric. Now how to check whether it is anti-symmetric or not. So for checking anti-symmetric we must satisfy the symmetric. Anti-symmetric is basically opposite of your symmetric. Now what we have in symmetric is that suppose if you have 2, 1 in a set then it is mandatory or it is necessary to have 1, 2 in that particular relation, right. So this is the criteria how you satisfy the condition for symmetric. Now exactly inverse we have for the anti-symmetric. Now to check whether it is anti-symmetric or not what we have to do just check 2, 1. So if you are having 2, 1 it means you shouldn't have 1, 2, right. If you have 2, 1 it means you shouldn't have 1, 2. So this is your anti-symmetric. Now there is a difference between anti-symmetric and asymmetric. The thing is that anti-symmetric allows you to have your diagonal values. So if you have diagonal values then you can say this is bit liberal then you can say it is a anti-symmetric. Now let's check for this. We have 1, 1, 2, 2, 3, 3. So if you have 2, 1 we don't have any of these elements rather than this. So this is not an anti-symmetric it means you have no condition to check. So obviously this will be your anti-symmetric. We have no ordered pair where we have to check for the symmetric or anti-symmetric. So we can say it satisfies your anti-symmetric property. This is reflexive it is anti-symmetric and now the third condition is your transitive. Now how to check for transitivity? Transitivity satisfies suppose if you have element a comma b and you have b comma c it means you can traverse from a to c right? So like this. So here you can form a comma c. So if you get this condition or if you satisfy this property it means it is transitive. Okay now let's check for our relation. We have 1, 1, we have 2, 2, we have 3, 3. Here I find no element which will be checking for transitivity. It means I don't have to check for transitivity. So we have reflexive, anti-symmetric and transitive. Here I mean I am satisfying the three conditions. See when you have to check for this condition when you have that type of element in a relation if you don't have that element in a relation automatically we will say it is transitive and anti-symmetric. So now R2 was your reflexive, transitive, anti-symmetric means this is nothing but your poset. So I can say my R2 is poset. Let us take one more example. Now think and write. Can we say if anyone or two condition satisfies then it is a partial order? Is this possible if it is anti-symmetric and transitive and not reflexive? Can you say still that set or relation is partial order? Think for a while and then answer. Answer is no. It should satisfy all the three conditions that is transitive, reflexive and anti-symmetric. Now let us take a simple example. Show that the relation greater than or equal to is a partial ordering on set of integers. Now the solution is let Z be the set of all integers and you have a relation greater than or equals to. So for the first step or for the first condition we will see integer A is greater than or equals to A for every integer A. Obviously if this is the case it is reflexive. You can put up two values for A and B where you find two integers and the relation is A is in relation with B and B is in relation with A. It means A is greater than B and obviously B is greater than A and here we say A equals to B. So even if this condition satisfies it means it is anti-symmetric. Now when you satisfy the three condition obviously it will be poset. So let us check for the transitivity which is your last property that is let A, B and C be any integer let A be greater than or equals to B and same for the C. Now obviously I have explained you with the help of example it means A is also greater than or equals to C and here we satisfy the third important condition that is transitive. Since the relation is transitive reflexive and anti-symmetric it is said as a poset. Therefore Z, greater than or equals to is nothing but your poset. So in this session we have studied all about poset. Now here are some of the references which I have used during the content preparation of this session. Thank you so much.