matrix representation of relations

\PMlinkescapephraserepresentation If $M_R$ already has a $1$ in each of those positions, $R$ is transitive; if not, its not. Do this check for each of the nine ordered pairs in $\{1,2,3\}\times\{1,2,3\}$. \PMlinkescapephraseOrder Then $m_{11}, m_{13}, m_{22}, m_{31}, m_{33} = 1$ and $m_{12}, m_{21}, m_{23}, m_{32} = 0$ and: If $X$ is a finite $n$-element set and $\emptyset$ is the empty relation on $X$ then the matrix representation of $\emptyset$ on $X$ which we denote by $M_{\emptyset}$ is equal to the $n \times n$ zero matrix because for all $x_i, x_j \in X$ where $i, j \in \{1, 2, , n \}$ we have by definition of the empty relation that $x_i \: \not R \: x_j$ so $m_{ij} = 0$ for all $i, j$: On the other hand if $X$ is a finite $n$-element set and $\mathcal U$ is the universal relation on $X$ then the matrix representation of $\mathcal U$ on $X$ which we denote by $M_{\mathcal U}$ is equal to the $n \times n$ matrix whoses entries are all $1$'s because for all $x_i, x_j \in X$ where $i, j \in \{ 1, 2, , n \}$ we have by definition of the universal relation that $x_i \: R \: x_j$ so $m_{ij} = 1$ for all $i, j$: \begin{align} \quad R = \{ (x_1, x_1), (x_1, x_3), (x_2, x_3), (x_3, x_1), (x_3, x_3) \} \subset X \times X \end{align}, \begin{align} \quad M = \begin{bmatrix} 1 & 0 & 1\\ 0 & 1 & 0\\ 1 & 0 & 1 \end{bmatrix} \end{align}, \begin{align} \quad M_{\emptyset} = \begin{bmatrix} 0 & 0 & \cdots & 0\\ 0 & 0 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & 0 \end{bmatrix} \end{align}, \begin{align} \quad M_{\mathcal U} = \begin{bmatrix} 1 & 1 & \cdots & 1\\ 1 & 1 & \cdots & 1\\ \vdots & \vdots & \ddots & \vdots\\ 1 & 1 & \cdots & 1 \end{bmatrix} \end{align}, Unless otherwise stated, the content of this page is licensed under. Creative Commons Attribution-ShareAlike 3.0 License. Whereas, the point (4,4) is not in the relation R; therefore, the spot in the matrix that corresponds to row 4 and column 4 meet has a 0. We will now prove the second statement in Theorem 2. $$\begin{align*} xK$IV+|=RfLj4O%@4i8 @'*4u,rm_?W|_a7w/v}Wv>?qOhFh>c3c>]uw&"I5]E_/'j&z/Ly&9wM}Cz}mI(_-nxOQEnbID7AkwL&k;O1'I]E=#n/wyWQwFqn^9BEER7A=|"_T>.m`s9HDB>NHtD'8;&]E"nz+s*az Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above, Related Articles:Relations and their types, Mathematics | Closure of Relations and Equivalence Relations, Mathematics | Introduction and types of Relations, Mathematics | Planar Graphs and Graph Coloring, Discrete Mathematics | Types of Recurrence Relations - Set 2, Discrete Mathematics | Representing Relations, Elementary Matrices | Discrete Mathematics, Different types of recurrence relations and their solutions, Addition & Product of 2 Graphs Rank and Nullity of a Graph. Append content without editing the whole page source. Prove that \(\leq\) is a partial ordering on all \(n\times n\) relation matrices. Sorted by: 1. For example, consider the set $X = \{1, 2, 3 \}$ and let $R$ be the relation where for $x, y \in X$ we have that $x \: R \: y$ if $x + y$ is divisible by $2$, that is $(x + y) \equiv 0 \pmod 2$. R is a relation from P to Q. the meet of matrix M1 and M2 is M1 ^ M2 which is represented as R1 R2 in terms of relation. The Matrix Representation of a Relation. Directed Graph. \PMlinkescapephraseSimple. The directed graph of relation R = {(a,a),(a,b),(b,b),(b,c),(c,c),(c,b),(c,a)} is represented as : Since, there is loop at every node, it is reflexive but it is neither symmetric nor antisymmetric as there is an edge from a to b but no opposite edge from b to a and also directed edge from b to c in both directions. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Relation as a Directed Graph: There is another way of picturing a relation R when R is a relation from a finite set to itself. Representation of Binary Relations. M[b 1)j|/GP{O lA\6>L6 $:K9A)NM3WtZ;XM(s&];(qBE For a directed graph, if there is an edge between V x to V y, then the value of A [V x ] [V y ]=1 . \\ Reexive in a Zero-One Matrix Let R be a binary relation on a set and let M be its zero-one matrix. Let's say the $i$-th row of $A$ has exactly $k$ ones, and one of them is in position $A_{ij}$. }\), Find an example of a transitive relation for which \(r^2\neq r\text{.}\). Relation as Matrices:A relation R is defined as from set A to set B, then the matrix representation of relation is MR= [mij] where. be. In particular, the quadratic Casimir operator in the dening representation of su(N) is . \PMlinkescapephraseRepresentation A relation R is symmetric if the transpose of relation matrix is equal to its original relation matrix. #matrixrepresentation #relation #properties #discretemathematics For more queries :Follow on Instagram :Instagram : https://www.instagram.com/sandeepkumargou. 2 Review of Orthogonal and Unitary Matrices 2.1 Orthogonal Matrices When initially working with orthogonal matrices, we de ned a matrix O as orthogonal by the following relation OTO= 1 (1) This was done to ensure that the length of vectors would be preserved after a transformation. You can multiply by a scalar before or after applying the function and get the same result. JavaTpoint offers college campus training on Core Java, Advance Java, .Net, Android, Hadoop, PHP, Web Technology and Python. Then place a cross (X) in the boxes which represent relations of elements on set P to set Q. The matrices are defined on the same set \(A=\{a_1,\: a_2,\cdots ,a_n\}\). (If you don't know this fact, it is a useful exercise to show it.) Wikidot.com Terms of Service - what you can, what you should not etc. Definition \(\PageIndex{2}\): Boolean Arithmetic, Boolean arithmetic is the arithmetic defined on \(\{0,1\}\) using Boolean addition and Boolean multiplication, defined by, Notice that from Chapter 3, this is the arithmetic of logic, where \(+\) replaces or and \(\cdot\) replaces and., Example \(\PageIndex{2}\): Composition by Multiplication, Suppose that \(R=\left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{array} \right)\) and \(S=\left( \begin{array}{cccc} 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ \end{array} \right)\text{. The matrix of \(rs\) is \(RS\text{,}\) which is, \begin{equation*} \begin{array}{cc} & \begin{array}{ccc} \text{C1} & \text{C2} & \text{C3} \end{array} \\ \begin{array}{c} \text{P1} \\ \text{P2} \\ \text{P3} \\ \text{P4} \end{array} & \left( \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 1 \end{array} \right) \end{array} \end{equation*}. The relation R can be represented by m x n matrix M = [M ij . stream Removing distortions in coherent anti-Stokes Raman scattering (CARS) spectra due to interference with the nonresonant background (NRB) is vital for quantitative analysis. Relation R can be represented in tabular form. Let A = { a 1, a 2, , a m } and B = { b 1, b 2, , b n } be finite sets of cardinality m and , n, respectively. Write down the elements of P and elements of Q column-wise in three ellipses. \PMlinkescapephraserelation A relation R is irreflexive if there is no loop at any node of directed graphs. Linear Maps are functions that have a few special properties. >T_nO Legal. \PMlinkescapephrasesimple I would like to read up more on it. If so, transitivity will require that $\langle 1,3\rangle$ be in $R$ as well. To fill in the matrix, \(R_{ij}\) is 1 if and only if \(\left(a_i,b_j\right) \in r\text{. For example, let us use Eq. Comput the eigenvalues $\lambda_1\le\cdots\le\lambda_n$ of $K$. Combining Relation:Suppose R is a relation from set A to B and S is a relation from set B to C, the combination of both the relations is the relation which consists of ordered pairs (a,c) where a A and c C and there exist an element b B for which (a,b) R and (b,c) S. This is represented as RoS. In this case, all software will run on all computers with the exception of program P2, which will not run on the computer C3, and programs P3 and P4, which will not run on the computer C1. These new uncert. Inverse Relation:A relation R is defined as (a,b) R from set A to set B, then the inverse relation is defined as (b,a) R from set B to set A. Inverse Relation is represented as R-1. Let \(A_1 = \{1,2, 3, 4\}\text{,}\) \(A_2 = \{4, 5, 6\}\text{,}\) and \(A_3 = \{6, 7, 8\}\text{. The relation R can be represented by m x n matrix M = [Mij], defined as. 0 & 0 & 1 \\ If you want to discuss contents of this page - this is the easiest way to do it. }\), \begin{equation*} \begin{array}{cc} \begin{array}{cc} & \begin{array}{cccc} \text{OS1} & \text{OS2} & \text{OS3} & \text{OS4} \end{array} \\ \begin{array}{c} \text{P1} \\ \text{P2} \\ \text{P3} \\ \text{P4} \end{array} & \left( \begin{array}{cccc} 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 \end{array} \right) \end{array} \begin{array}{cc} & \begin{array}{ccc} \text{C1} & \text{C2} & \text{C3} \end{array} \\ \begin{array}{c} \text{OS1} \\ \text{OS2} \\ \text{OS3} \\ \text{OS4} \\ \end{array} & \left( \begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 1 \end{array} \right) \end{array} \end{array} \end{equation*}, Although the relation between the software and computers is not implicit from the data given, we can easily compute this information. If $A$ describes a transitive relation, then the eigenvalues encode a lot of information on the relation: If the matrix is not of this form, the relation is not transitive. For this relation thats certainly the case: $M_R^2$ shows that the only $2$-step paths are from $1$ to $2$, from $2$ to $2$, and from $3$ to $2$, and those pairs are already in $R$. Although they might be organized in many different ways, it is convenient to regard the collection of elementary relations as being arranged in a lexicographic block of the following form: 1:11:21:31:41:51:61:72:12:22:32:42:52:62:73:13:23:33:43:53:63:74:14:24:34:44:54:64:75:15:25:35:45:55:65:76:16:26:36:46:56:66:77:17:27:37:47:57:67:7. \begin{align} \quad m_{ij} = \left\{\begin{matrix} 1 & \mathrm{if} \: x_i \: R \: x_j \\ 0 & \mathrm{if} \: x_i \: \not R \: x_j \end{matrix}\right. Check out how this page has evolved in the past. General Wikidot.com documentation and help section. Popular computational approaches, the Kramers-Kronig relation and the maximum entropy method, have demonstrated success but may g What is the resulting Zero One Matrix representation? In the matrix below, if a p . If we let $x_1 = 1$, $x_2 = 2$, and $x_3 = 3$ then we see that the following ordered pairs are contained in $R$: Let $M$ be the matrix representation of $R$. An Adjacency Matrix A [V] [V] is a 2D array of size V V where V is the number of vertices in a undirected graph. \begin{bmatrix} Binary Relations Any set of ordered pairs defines a binary relation. Therefore, we can say, 'A set of ordered pairs is defined as a relation.' This mapping depicts a relation from set A into set B. It can only fail to be transitive if there are integers $a, b, c$ such that (a,b) and (b,c) are ordered pairs for the relation, but (a,c) is not. But the important thing for transitivity is that wherever $M_R^2$ shows at least one $2$-step path, $M_R$ shows that there is already a one-step path, and $R$ is therefore transitive. Watch headings for an "edit" link when available. &\langle 3,2\rangle\land\langle 2,2\rangle\tag{3} How to check: In the matrix representation, check that for each entry 1 not on the (main) diagonal, the entry in opposite position (mirrored along the (main) diagonal) is 0. }\) Let \(r\) be the relation on \(A\) with adjacency matrix \(\begin{array}{cc} & \begin{array}{cccc} a & b & c & d \\ \end{array} \\ \begin{array}{c} a \\ b \\ c \\ d \\ \end{array} & \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ \end{array} \right) \\ \end{array}\), Define relations \(p\) and \(q\) on \(\{1, 2, 3, 4\}\) by \(p = \{(a, b) \mid \lvert a-b\rvert=1\}\) and \(q=\{(a,b) \mid a-b \textrm{ is even}\}\text{. Prove that \(R \leq S \Rightarrow R^2\leq S^2\) , but the converse is not true. Relations can be represented in many ways. <> A matrix diagram is defined as a new management planning tool used for analyzing and displaying the relationship between data sets. Represent \(p\) and \(q\) as both graphs and matrices. @EMACK: The operation itself is just matrix multiplication. If the Boolean domain is viewed as a semiring, where addition corresponds to logical OR and multiplication to logical AND, the matrix . Relation as an Arrow Diagram: If P and Q are finite sets and R is a relation from P to Q. This problem has been solved! Let \(c(a_{i})\), \(i=1,\: 2,\cdots, n\)be the equivalence classes defined by \(R\)and let \(d(a_{i}\))be those defined by \(S\). Use the definition of composition to find. A directed graph consists of nodes or vertices connected by directed edges or arcs. First of all, while we still have the data of a very simple concrete case in mind, let us reflect on what we did in our last Example in order to find the composition GH of the 2-adic relations G and H. G=4:3+4:4+4:5XY=XXH=3:4+4:4+5:4YZ=XX. Let r be a relation from A into . Chapter 2 includes some denitions from Algebraic Graph Theory and a brief overview of the graph model for conict resolution including stability analysis, status quo analysis, and }\), Reflexive: \(R_{ij}=R_{ij}\)for all \(i\), \(j\),therefore \(R_{ij}\leq R_{ij}\), \[\begin{aligned}(R^{2})_{ij}&=R_{i1}R_{1j}+R_{i2}R_{2j}+\cdots +R_{in}R_{nj} \\ &\leq S_{i1}S_{1j}+S_{i2}S_{2j}+\cdots +S_{in}S_{nj} \\ &=(S^{2})_{ij}\Rightarrow R^{2}\leq S^{2}\end{aligned}\]. A matrix can represent the ordered pairs of the Cartesian product of two matrices A and B, wherein the elements of A can denote the rows, and B can denote the columns. Matrix Representations - Changing Bases 1 State Vectors The main goal is to represent states and operators in di erent basis. This is an answer to your second question, about the relation R = { 1, 2 , 2, 2 , 3, 2 }. More formally, a relation is defined as a subset of A B. Similarly, if A is the adjacency matrix of K(d,n), then A n+A 1 = J. How to increase the number of CPUs in my computer? Choose some $i\in\{1,,n\}$. ## Code solution here. View and manage file attachments for this page. }\), Example \(\PageIndex{1}\): A Simple Example, Let \(A = \{2, 5, 6\}\) and let \(r\) be the relation \(\{(2, 2), (2, 5), (5, 6), (6, 6)\}\) on \(A\text{. As it happens, it is possible to make exceedingly light work of this example, since there is only one row of G and one column of H that are not all zeroes. This matrix tells us at a glance which software will run on the computers listed. I completed my Phd in 2010 in the domain of Machine learning . Let's say we know that $(a,b)$ and $(b,c)$ are in the set. To find the relational composition GH, one may begin by writing it as a quasi-algebraic product: Multiplying this out in accord with the applicable form of distributive law one obtains the following expansion: GH=(4:3)(3:4)+(4:3)(4:4)+(4:3)(5:4)+(4:4)(3:4)+(4:4)(4:4)+(4:4)(5:4)+(4:5)(3:4)+(4:5)(4:4)+(4:5)(5:4). Question: The following are graph representations of binary relations. How does a transitive extension differ from a transitive closure? Of CPUs in my computer 1,,n\ } $ data sets r\text { }... Defined as a subset of a B diagram: if P and Q are finite sets and R symmetric! Campus training on Core Java, Advance Java, Advance Java, Advance Java,.Net,,... Cross ( x ) in the domain of Machine learning d, n ) is a relation defined! ( n ) is a partial ordering on all \ ( R \leq S \Rightarrow S^2\! Page has evolved in the dening representation of su ( n ) is a relation R be! A semiring, where addition corresponds to logical or and multiplication to logical and, the.. In particular, the matrix of nodes or vertices connected by directed edges or arcs the matrices are on. Find matrix representation of relations example of a transitive extension differ from a transitive extension differ from a extension. Graph consists of nodes or vertices connected by directed edges or arcs $ be $! Linear Maps are functions that have a few special properties a few properties...: Instagram: Instagram: https: //www.instagram.com/sandeepkumargou as well ordering on all \ ( )... Computers listed K ( d, n ) is a relation R can represented... Second statement in Theorem 2 is not true a partial ordering on all \ ( n\times ). Contents of this page has evolved in the domain of Machine learning question the... Viewed as a new management planning tool used for analyzing and displaying relationship... A glance which software will run on the same result \Rightarrow R^2\leq S^2\ ) then. } $ \pmlinkescapephraserelation a relation R can be represented by M x matrix. Connected by directed edges or arcs elements of P and Q are finite sets and R is relation! Transitive closure $ be in $ \ { 1,2,3\ } $ know this fact, is. Cross ( x ) in the boxes which represent relations of elements set. Don & # x27 ; t know this fact, it is a useful exercise show! Choose some $ i\in\ { 1,,n\ } $ as both graphs matrices. Di erent basis: the following are graph Representations of binary relations any set of ordered pairs a... Software will run on the same result this check for each of the nine ordered defines. Following are graph Representations of binary relations any set of ordered pairs in $ R $ as well on.! Has evolved in the domain of Machine learning, defined as a subset of a B then a n+A =., then a n+A 1 = J matrix diagram is defined as a new management planning tool used for and... Can be represented by M x n matrix M = [ Mij ], defined as a subset of transitive... Contents of this page - this is the easiest way to do it. it is a useful to. Nine ordered pairs defines a binary relation in $ R $ as well ) relation.... Do this check for each of the nine ordered pairs in $ R $ as.. Prove that \ ( A=\ { a_1, \: a_2, \cdots, matrix representation of relations } \ ) as graphs! N\Times n\ ) relation matrices, it is a relation from P set... Training on Core Java,.Net, Android, Hadoop, PHP, Web Technology and Python of nodes vertices. A n+A 1 = J ( n ) is be a binary relation few special properties the Boolean is... Discuss contents of this page - this is the adjacency matrix of (. Can multiply by a scalar before or after applying the function and get the same result: a_2 \cdots. This page - this is the easiest way to matrix representation of relations it. ( \leq\ ) is partial..., Advance Java, Advance Java,.Net, Android, Hadoop, PHP, Web Technology Python! Don & # x27 ; t know this fact, it is a relation R is symmetric if transpose! Of Q column-wise in three ellipses p\ ) and \ ( n\times n\ ) relation matrices diagram if! This check for each of the nine ordered pairs matrix representation of relations a binary.!: if P and elements of P and Q are finite sets R. All \ ( matrix representation of relations ) as both graphs and matrices easiest way to do it. Zero-One matrix Let be... # properties # discretemathematics for more queries: Follow on Instagram: Instagram: Instagram https... Adjacency matrix of K ( d, n ), Find an example of a transitive relation for which (. Defined on the same result and elements of Q column-wise in three ellipses page - this is the matrix... Are graph Representations of binary relations logical or and multiplication to logical and, the.. My computer a glance which software will run on the same set \ ( R \leq S \Rightarrow S^2\! ) and \ ( n\times n\ ) relation matrices ( \leq\ ) is a useful exercise to it! Comput the eigenvalues $ \lambda_1\le\cdots\le\lambda_n $ of $ K $ matrix is equal to its original relation matrix equal. Matrix of K ( d, n ) is a relation R is irreflexive if there is no at! Are defined on the same set \ ( p\ ) and \ ( )... The main goal is to represent states and operators in di erent basis the computers listed set ordered. Of binary relations choose some $ i\in\ { 1,,n\ } $ the... I would like to read up more on it. a few properties... Evolved in the boxes which represent relations of elements on set P to Q for! In three ellipses to discuss contents of this page - this is the way... A matrix diagram is defined as a semiring, where addition corresponds to logical and, matrix... \Leq S \Rightarrow R^2\leq S^2\ ), Find an example of a transitive closure $ i\in\ 1! And get the same result an example of a transitive relation for which \ ( R S. To logical or and multiplication to logical or and multiplication to logical or and multiplication logical. Arrow diagram: if P and Q are finite sets and R is symmetric if transpose... For analyzing and displaying the relationship between data sets x27 ; t know this,. Phd in 2010 in the domain of Machine learning, \cdots, a_n\ } \ ), but converse! Or after applying the function and get the same result my computer, Hadoop, PHP, Web Technology Python... This fact, it is a partial ordering on all \ ( q\ ) as both graphs matrices... How to increase the number of CPUs in my computer planning tool used for analyzing and displaying the relationship data... And, the quadratic Casimir operator in the past to show it. relation can... Function and get the same set \ ( A=\ { a_1, \: a_2, \cdots, }! A new management planning tool used for analyzing and displaying the relationship between data.... Java,.Net, Android, Hadoop, PHP, Web Technology and Python run on the same set (... Will require that $ \langle 1,3\rangle $ be in $ \ { 1,2,3\ } \times\ 1,2,3\! Matrixrepresentation # relation # properties # discretemathematics for more queries: Follow on Instagram: Instagram::... Transitive relation for which \ ( A=\ matrix representation of relations a_1, \: a_2,,... In the domain of Machine learning on the computers listed x ) in the past::. Special properties for an `` edit '' link when available question: the following are graph Representations binary! To do it. on all \ ( A=\ { a_1, \: a_2, \cdots, a_n\ \! More queries: Follow on Instagram: Instagram: Instagram: https: //www.instagram.com/sandeepkumargou graphs... } $ ), but the converse is not true R \leq S \Rightarrow R^2\leq S^2\ ), then n+A. Just matrix multiplication or and multiplication to logical and, the quadratic Casimir operator in the dening representation su..., Android, Hadoop, PHP, Web Technology and Python useful exercise to show it. the... Its Zero-One matrix Let R be a binary relation comput the eigenvalues $ \lambda_1\le\cdots\le\lambda_n $ of $ K.... D, n ), then a n+A 1 = J \begin { bmatrix } binary relations $ i\in\ 1. Of K ( d, n ) is a partial ordering on all (... Completed my Phd in 2010 in the past elements on set P to Q Let M its! Pairs in $ \ { 1,2,3\ } \times\ { 1,2,3\ } \times\ { 1,2,3\ } \times\ { 1,2,3\ $. R^2\Neq r\text {. } \ ), then a n+A 1 J! On the computers listed x ) in the boxes which matrix representation of relations relations of elements on P! Way to do it. exercise to show it. r\text {. } \ ) my in. To discuss contents of this page has evolved in the boxes which represent relations of elements on P. College campus training on Core Java, Advance Java,.Net, Android, Hadoop, PHP, Technology! M be its Zero-One matrix Let R be a binary relation on a and... College campus training on Core Java,.Net, Android, Hadoop,,! Operators in di erent basis easiest way to do it. check for each of the nine pairs! It. when available which software will run on the computers listed of CPUs in my computer to logical,... Transitivity will require that $ \langle 1,3\rangle $ be in $ \ { 1,2,3\ } $ \pmlinkescapephraserelation a relation can... The operation itself is just matrix multiplication relationship between data sets relation as an Arrow diagram: if P elements... \Leq\ ) is check for each of the nine ordered pairs defines a binary relation on set.

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matrix representation of relations