 Hello everyone, I am Lachna Pathak from Valchan Institute of Technology, Sholapur. In this video session, we are going to see all about relation matrix and graph. Learning outcome. At the end of this session, student will be able to evaluate relation matrix and graphs on it. So we will see all about how we can draw matrix and graphs based on a particular relation. Now what is a relation matrix? A relation R from a finite set A to a finite set B can be represented by a matrix called the relation matrix of R. Now when I explain or when I talk about matrix, it means there are two finite set. Two finite set A and B. Now I can relate set A with set B with some type of relation and that is nothing but a relation between two set. Now when I talk about matrix, I mean your matrix is nothing but tabular representation between your two relation. Now let us take a simple example. Let A equal to be A1, A2, A3 till AM and B equals to be B1, B2, B3 till Bn. So basically these are some elements in set A1 to AM are elements of set A, B1 to Bn are elements of set B. Now here these sets are finite sets. Now they contain M and N element respectively and R be the relation from set A to B. R is nothing but the relation between set A to B. Now they can be represented by M into N matrix and we can use term MR, capital M with a subscript R and the element can be represented by IJ. R will be an element, I and J is nothing but indexed. Now let us see how mathematically these things are represented. So an element R, I, J is nothing but row and column. These are indexed for a particular set which starts from element 1. That means if in a matrix, when you represent in a tabular format, if it is 1, it means the element is present and rest all element will be 0. So basically matrix is represented by 0 and 1 element. Now this is possible for A, I which is in relation with B, J. Let us see and for 0 if it is A, I related to B, J. Example let A equal to be 1, 2, 3, 4 and B matrix be B1, B2 and B3. So these are the elements in set A and set B. Now consider the relation R where you have 1, B2, 1, B3, 3, B2, 4, B1, 4, B3. So this is nothing but a relation between set A to set B. Now my question is determine the matrix of the relation. So let us see how we determine it. Initially we have set A, so in the solution set I will write the same thing. In set A we have 1, 2, 3, 4 and in set B we have B1, B2, B3 and the relation R was 1, B2, 1, B3, 3, B3 and all this. Now matrix of the relation R is written as in this format. So the set B was B1, B2, B3. So it is M into N. So N is nothing but my column and M is nothing but my rows. So these are my rows that is set A, 1, 2, 3, 4 and set B is B1, B2, B3. So this is how we represent in a matrix format. Now I will wipe out set A and B. So this is how it looks like. Now my question is let A be a set 1, 2, 3, 4, find the relation R on A determined by the matrix. Take a pause, think on this and write down the relation or matrix. So I will consider all the element which has the value 1 in my matrix. So it will be 1, B1, 3, B1. So here you can see in the solution set 1, 1, 1, 3, 2, 3, 3, 1, 4, 1, 4, 2, 4, 4. Instead of B1 I have taken as 1, 2, 3. So this is all about matrix. Now let us see what are properties of a relation in a set. First you have reflexive. Reflexive is nothing but where all diagonal entries must be 1. Next property is symmetric. Symmetric is nothing but where you have R, I, J equals to R, I, J for every I and J. It means both the elements should be equal. Let us say in more detail with an example, anti-symmetry. Anti-symmetry is nothing but R, I, J equals to 1 and R, J, I will be equals to 0. But the condition is for I is not equals to J. It means we allow this in a diagonal manner. I will show you with an example. See in case of reflexive we have all the diagonal element as 1. So if in a given matrix all your diagonal element are 1, it means your matrix is reflexive. Here we see R, I, J equals to R, I, J. It means m of 1, 1 will be equal to 1, 1. This is nothing but your reflexive. Now for symmetric let us consider first two examples. Now this is R, I, J. It means 1, 1. The element 1, 1 is 1. It means the inverse that is 1, 1 will be 1. Now let us take another example. Now let us take this example. So it is 2, 1. 2, 1 is 1. It means 1, 2 should also be 1. And if this is the case it is said to be symmetric. Now let us check 2, 1 is 1. It means 1, 2 is also 1. So this matrix is said as symmetric matrix. As I already told you that you will get more clear example. Now let us move to anti-symmetric. In case of anti-symmetric it is just similar to your symmetric but the only thing is here we allow diagonal entries. Obviously for case of anti-symmetric 1, 1 is 1. Obviously I told you if it is having diagonal entries we allow. Now let us check for 2, 1. 2, 1 is 1. It means 1, 2 should be 0. So 2, 1 is 1. Now see 1, 2 is 0. If this is the case it means it is anti-symmetric. Now let us move to graph. Now see let R be a relation in a finite set A. It means A1, A2, An. An element of A are represented by points or circles called nodes. So this is basically the description of your graph has been represented. These nodes are nothing but called as vertices. These are used to show the connection between your edges. Now let us see a simple example of your graph. Here we see your vertex A is connected to vertex B but the relation is from A to B. Another graph is here we see A is connected with A. It means it is nothing but a loop. So A is in relation with A and here we see A is in relation with B. Now of course this is directed graph. See here we see one more example, A is in relation with B and B is in relation with B. So A is related with B and B in relation with B. Here we see one more example, A is in relation with B and B is in relation with C and C is in relation with A. So let us take a simple example. Now let A equal to be 5, 6, 7, 8 and R equal to be x, y such that your x is greater than y. Now let us draw the graph of R and also give its matrix. So now first of all I need a set. So the condition is x, y should be in such a condition that x is greater than y. So let us begin with the greatest element. So the set will be 8, 7, it will be 8, 6, 8, 5, right, 8, 7, 8, 6, 8, 5. Next will be 7, 7, 6, I will take 7, 5, another is 6, 5, right. So this is nothing but set. So here I have formed the relation between this set where my condition is x is greater than y. Now I will draw a graph and followed by a matrix. So here I have 5, 6, 7, 8. So let us draw. So 8 is in relation with 5, right. 8 is also in relation with 7, 8 is in relation with 6, now 7 is in relation with 6 and 7 is also in relation with 5, right. Now 6 is in relation with 5. It means the lower element are connected to the higher element. So this is how we draw the matrix, graph. Now with the help of same thing we can draw the matrix. So 5, 6, 7, 8, 5, 6, 7, 8, here we will have 4 by 4 matrix and you have to just fill up. So it is 8, 7, so 8, 7. So here I will fill up with 1, 8, 6, 8, 6, so here I will fill up with 1, 8, 5 and rest all will be 0. So this is how we draw matrix and graph for a particular relation. Thank you.