14/09/2015 Reminder About Basic Properties of Relations Definitions: VCU, Department of Computer Science A relation R on a set A is called reflexive if (a, a)R for every element aA. CMSC 302 Relations 2 A relation R on a set A is called symmetric if (b, a)R Vojislav Kecman whenever (a, b)R for all a, bA. A relation R on a set A is called transitive if whenever (a, b)R and (b, c)R, then (a, c)R for a, b, c A. 14/09/2015 1/57 14/09/2015 2/57 Representing Relations Representing Relations •Example: How can we represent the relation R defined •We already know different ways of representing relations. We will between the set A{1, 2, 3} and set B{1, 2} where now take a closer look at two ways of representation: Zero-one R = {(2, 1), (3, 1), (3, 2)} as a zero-one matrix? matrices and directed graphs (digraphs). •If R is a relation from A = {a1, a2, …, am} to B = •Solution: The matrix MR is given by {b1, b2, …, bn}, then R can be represented by the zero-one matrix MR = [mij] with •mij = 1, if (ai, bj)R, and 00 •mij = 0, if (ai, bj)R. M 10 R •Note that for creating this matrix we first need to list the elements 11 in A and B in a particular, but arbitrary order. 14/09/2015 3/57 14/09/2015 4/57 1 14/09/2015 Representing Relations Representing Relations •What do we know about the matrices representing a relation •What do we know about the matrices representing symmetric on a set (a relation from A to A) ? relations? t •They are square matrices. •These matrices are symmetric, that is, MR = (MR) . •What do we know about matrices representing reflexive relations? 1011 110 0 •All the elements on the diagonal of such matrices Mref must be 1s. 0100 110 0 M R M R 1 1001 110 0 1 1011 110 0 . M ref . •symmetric matrix, •non-symmetric matrix, . symmetric relation. non-symmetric relation. 1 14/09/2015 5/57 14/09/2015 6/57 Representing Relations Zero-One Reflexive, Symmetric •Terms:Reflexive, non-reflexive, irreflexive, symmetric, asymmetric, and antisymmetric. •The Boolean operations join and meet can be used to – These relation characteristics are very easy to determine the matrices representing the union and the recognize by inspection of the zero-one matrix. intersection of two relations, respectively. 1 0 1 0 •To obtain the join of two zero-one matrices, we apply the any- any- Boolean “or” function to all corresponding elements in the a thing thing a n n yt 1 0 yt 0 h matrices. 1 h in 1 in 1 g g 1 0 any- any- •To obtain the meet of two zero-one matrices, we apply the thing 1 thing 0 0 0 Boolean “and” function to all corresponding elements in the matrices. Symmetric: Antisymmetric: Reflexive: Irreflexive: all identical all 1’s are across all 1’s on diagonal all 0’s on diagonal across diagonal from 0’s 14/09/2015 7/57 14/09/2015 8/57 2 14/09/2015 Representing Relations Using Matrices Representing Relations Using Matrices •Example: Let the relations R and S be represented by the matrices -Do you remember the Boolean product of two zero-one 101 101 matrices? M 100 M 011 R S -Let A = [aij] be an mk zero-one matrix and 010 100 B = [bij] be a kn zero-one matrix. •What are the matrices representing RS and RS? -Then the Boolean product of A and B, denoted by AB, is •Solution: These matrices are given by the mn matrix with (i, j)th entry [cij], where -cij = (ai1 b1j) (ai2 b2i) … (aik bkj). 101 101 -cij = 1 if and only if at least one of the terms MMM111 MMMRS R S 000 RS R S (a b ) = 1 for some n; otherwise c = 0. 000 in nj ij 110 14/09/2015 9/57 14/09/2015 10/57 Representing Relations Using Matrices Representing Relations Using Matrices -Let us now assume that the zero-one matrices -This gives us the following rule: MA = [aij], MB = [bij] and MC = [cij] represent relations A, B, and C, respectively. -MBA = MAMB -Remember: For MC = MAMB we have: -In other words, the matrix representing the composite of relations A and B is the Boolean product of the matrices -cij = 1 if and only if at least one of the terms representing A and B. (ain bnj) = 1 for some n; otherwise cij = 0. -In terms of the relations, this means that C contains a pair (x , i -Analogously, we can find matrices representing the powers z ) if and only if there is an element y such that (x , y ) is in j n i n of relations: relation A and [n] (yn, zj) is in relation B. -MRn = MR (n-th Boolean power). -Therefore, C = BA (composite of A and B). 14/09/2015 11/57 14/09/2015 12/57 3 14/09/2015 Representing Relations Using Matrices Representing Relations Using Digraphs •Example: Find the matrix representing R2, where the matrix representing R is given by •Solution: The matrix for R2 is given by 010 •Definition: A directed graph, or digraph, consists of a set V M 011 of vertices (or nodes) together with a set E of ordered pairs of R 011 elements of V called edges (or arcs). 100 [2] MM2 111 R R •The vertex a is called the initial vertex of the edge (a, b), and 010 the vertex b is called the terminal vertex of this edge. How did we get this one? Well, … •We can use arrows to display graphs. 010 010 (00)(10)(01) 1 1 011 011 1 (01)(11)(10) 1 100 100 0 1 (10)(01)(00) 14/09/2015 13/57 14/09/2015 14/57 Representing Relations Using Digraphs Using Directed Graphs •Example: Display the digraph with V = {a, b, c, d}, E = {(a, b), (a, d), (b, b), (b, d), (c, a), (c, b), (d, b)}. •Def.A directed graph or digraph G=(VG,EG) is a set VG of vertices (nodes) with a set EGVG×VG of edges a (arcs,links). Visually represented using dots for nodes, b and arrows for edges. Notice that a relation R:A×B can be represented as a graph GR=(VG=AB, EG=R). Edge set EG Matrix representation MR: Graph representation GR: (blue arrows) Susan Mary Sally Joe Susan d c Joe 1 1 0 Fred Mary Fred 0 1 0 Mark 0 0 1 Mark Sally •An edge of the form (b, b) is called a loop. Node set VG (black dots) 14/09/2015 15/57 14/09/2015 16/57 4 14/09/2015 Digraph Reflexive, Symmetric Representing Relations Using Digraphs • It is extremely easy to recognize the reflexive/irreflexive/ symmetric/antisymmetric •Obviously, we can represent any relation R on a set A by the properties by graph inspection. digraph with A as its vertices and all pairs (a, b)R as its edges. •Vice versa, any digraph with vertices V and edges E can be represented by a relation on V containing all the pairs in E. •This one-to-one correspondence between relations and digraphs means that any statement about relations also applies Reflexive: Irreflexive: Symmetric: Antisymmetric: to digraphs, and vice versa. Every node No node Every link is No link is has a self-loop links to itself bidirectional bidirectional This then means that digraphs are sets, and that all the set These are asymmetric & non-antisymmetric These are non-reflexive & non-irreflexive operations apply. We’ll use it in closures which come next! 14/09/2015 17/57 14/09/2015 18/57 Closures of Relations, or Closures of Relations Relational Closures •Def.For any property X, the “X closure” of a set A is defined as the “smallest” superset of A that has the given property. • Three types we will study The reflexive closure of a relation R on A is obtained by adding (a, a) to R for each aA. i.e.,it is R IA – Reflexive The symmetric closure of R is obtained by adding (b,a) to R •Easy for each (a, b) in R. i.e., it is R RT(note in book is R-1 used) –Symmetric • The transitive closure or connectivity relation of R is obtained by repeatedly adding (a,c) to R for each (a,b),(b,c) in •Easy R., i.e., it is * n 23nn 1 –Transitive R R R RR R R •Hard nZ 14/09/2015 19/57 14/09/2015 20/57 5 14/09/2015 Reflexive closure Reflexive closure • Consider a relation R: • In order to find the reflexive closure of a – Note that it is not reflexive g relation R, we add a loop at each node • We want to add edges to that does not have one make the relation reflexive f c d • By adding those edges, e • The reflexive closure of R is R U we have made a non- b reflexive relation R into –Where = { (a, a) | a R } a reflexive relation • Called the “diagonal relation” a – With matrices, we set the diagonal to all 1’s, • This new relation is called the reflexive closure of R meaning = diag([1 1 1 … 1]) 14/09/2015 21/57 14/09/2015 22/57 Reflexive closure example Reflexive closure example with matrices • Let R be a relation on the set { 0, 1, 2, 3 } containing the • Let R be a relation on the set { 0, 1, 2, 3 } containing the ordered pairs ordered pairs (0,1), (1,1), (1,2), (2,0), (2,2), and (3,0) (0,1), (1,1), (1,2), (2,0), (2,2), and (3,0) • What is the reflexive closure of R? • What is the reflexive closure of R? • We add all pairs of edges (a,a) that do not already exist • We ‘add’ a diagonal matrix with ones, called also identity matrix 0100 10001 100 01 0110 0100 0110 We add edges: R (0,0), (3,3) 1010 0010 1010 1000 0001 1001 32 We add entries: (0,0), (3,3) 14/09/2015 23/57 14/09/2015 24/57 6 14/09/2015 Symmetric closure Symmetric closure • Consider a relation R: • In order to find the symmetric closure of a – Note that it is not symmetric g relation R, we add an edge from a to b, • We want to add edges to where there is already an edge from b to a make the relation symmetric f c d T • By adding those edges, e • The symmetric closure of R is R U R we have made a non- b symmetric relation R into –If R = { (a,b) | … } a symmetric relation – Then RT = { (b,a) | … } a • This new relation is called the symmetric closure of R 14/09/2015 25/57 14/09/2015 26/57 Symmetric closure example Symmetric closure example with matrices • Let R