 from Walchand Institute of Technology, Sholapur and today in this video session we are going to learn all about haze diagrams. So what will be the learning outcome? At the end of this session student will be able to illustrate the haze diagram. Basically we will have discussion on what exactly is haze diagram. Why do we convert, post it into your haze diagram and how shall we do that. Now what are the prerequisite for these? Basically you should be aware about discrete mathematics, basics of your relation and the most important thing is what is a partial order relation. We are already aware partial order relation satisfies the basic three properties like reflexivity, transitivity and anti-symmetric which further sees the relation is partial order relation. Now let us see history. Haze diagrams are named after helmet haze. They are called as haze diagram because of the effective use haze made for them. Haze diagrams were originally devised as a technique for making drawings of partially ordered set that is nothing but named as post sets by hand and they have more recently been created automatically using graph drawing techniques. Now the phrase haze diagram may also refer to the transitive reduction as an abstract directed or cyclic graph independently of any drawing of that graph. Now why haze diagram? Now we see the main reason for converting post set into haze diagram is to reduce the complexity. The traditional approach is to gradually or we say the traditional approach is to form the tabular method likewise we do in set but when we come to haze diagram it is complicated to follow this. So in simple terms we try to reduce the complexity by using different way or approach of representation. And we all are aware that using diagrammatic representation is always more easier than tabular or documenting to you know. Main purpose of this is to read partial order relation into more comfortable format and make easy to understand. Now of course haze diagrams are only for partial order relations. You cannot design haze diagram for any relation rather than your partial orders. Now beginning with the introduction a good haze diagram. Now what is a good haze diagram? Although haze diagrams are simple as well as more intuitive tool for dealing with finite post sets it turns out to be rather difficult to draw a good haze diagram. Now the reason is that there will be many possibilities or many possible ways to draw a haze diagram for a given post set. The simple technique of just starting with the minimal element of an order and then drawing greater element in incrementing order or form often produces a quite poor result. Symmetries and internal structure of the order are easily lost. So we have to take care not to lose any internal structures and draw without touching them. Now let us see how to make this possible. Diagrammatic representation. Haze diagram is a type of mathematical diagram used to represent a finite partially ordered set in the form of a drawing of its transitive reduction. Now for any partial order set suppose say s comma less than or equals to represents each element of s as a vertex in the plane and the relation here is less than or equals to. So you can see the relation is less than or equals to. Here we draw a line segment or curve that goes upward from x to y whenever your y covers x that is whenever x is less than y and there is no z such that x is less than z which is less than y. Now at times of representation curves may cross each other but must not touch any vertices other than their end points this is another important condition you should note. Now let us see example. So draw the Haze diagram representing the partial ordering and the question is a comma b in a set such that a divides b on 1, 2, 4, 6, 8, 12. Now let us move to the solution. Initially let us consider the same set let p be equals to 1, 2, 4, 6, 8, 12. Now here we see the partial order relation is less than or equals to. It means the x element are less than or equal to y elements and the condition is a divides on b it means x divides on y. So the relation is less than or equals to. Now let us see how the set is been generated. So according to this there are many possibilities. Now let us see example 1, 2. Now here you can see in 1, 2, 2 is divisible by 1 yes so if it is 2, 1 and 1 is less than 2. So you see 1, 2 will be a part of this set. So here we see 2 is divisible by 1 that means your x divides on y right so this condition has been satisfied and 2, 1 is less than 2 that means your 2 is greater than 1. Now next we see 1, 3. So 3 is divisible by 1 right your 3 is divisible by 1 yes and 3 is greater than 1 that means 1 is less than 3 it means your x element is less than your y element. So yes this can further check all the possible conditions like 1, 4 so 4 is divisible by 1 and your 1 is less than 4 that means your x element is less than your y element. Let us move 1, 6 we see your 6 is divisible by 1 and 1 is less than 6 that means x divides y so these are all the possible sets. Now let us see have to draw the diagram so our initial condition is I will take 1 right so after 1 you can see next element was 2 so 2 is divisible by 1 so I will connect 1 line and then you can see 3 is divisible by 1 but your 1 thing we notice is that your 3 cannot be divided by 2 so you cannot connect in this manner right. So you have one node we will move into incremental pattern so 2 divided by 1 2 is less than 1 is less than 2 next 3 by 1. Now let us see another element we have 4 yes I will increment it over here it means 4 is divisible by 2 of course one thing you notice over here is 4 is also divisible by 1 so it is growing in upward direction so 4 is divisible by 2 which is also divisible by 1. Now next element is 6, 6 can be divided by 3 so 6 divided by 3 and 6 can also be divided by 1 so I grow it in this incremental form. Now next element is 8 here we see 8 can be divided by 4 it can also be divided by 2 and 8 can also be divided by 1. Now you can check it 8 cannot be divided by 6 neither your 8 can be divided by your 3 so you cannot connect this vertices right. Now next element was 12 so 12 can be divided by 6 and your 12 can be divided by your 3 as well as your 1 so this is how we draw our poset diagram. Now let us see further see 12 now there is one more possibility that your 12 is also divisible by 4 so you connect it in this way another possibility is your 6 can also be divided by 2 so we connect these vertices with the help of curve so this is how we represent our poset diagram or we say haze diagram basically it is a poset but the way of representation is nothing but your haze diagram. So this was all about your haze diagram now I have given you one more example you can use this example for practicing think and write. Now the question arises can we draw more than one possible haze diagram it means for a particular problem can there be more than one haze diagram take a pause and think on it. The answer is yes of course you can draw multiple haze diagram for a given problem it depends upon your user but at the end all the haze diagram will be similar. So in this session we have studied all about the haze diagram these are some of the references which I have used during the preparation of these video thank you.