Representing Relations Using Digraphs Definition: A directed graph, or digraph, consists of a set Vof vertices (or nodes) together with a set Eof ordered pairs of elements of Vcalled edges (or arcs).The vertex ais called the initial vertex of the edge (a,b), and the vertex bis called the terminal vertex of this edge. nodes) together with a set E of ordered pairs of elements of V called . Representing Relations Using Digraphs ... •Relations are sets, and therefore, we can apply the usual set operations to them. Loops are fundamentally dull, so for the most part, we ignore them. initial vertex . a. is called the . edges (or . of the edge (a,b), and the vertex . arcs). … •If we have two relations R 1 and R 2, and both of them are from a set A to a set B, then we can combine them to R 1 vertices (or . An edge of the form (a,a) is called a loop. • The vertex a is called the initial vertex of the edge (a, b), and the vertex b … 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 Є A and c Є C and there exist an element b Є B for which (a,b) Є R and (b,c) Є S. A loop in a loop-digraph is an arc from a vertex v to v: So loops are techincally arcs of the form (v;v) and look like loops when drawn. A digraph D is a loop-digraph … now, either that is an arc from v to w in the digraph, or there isn’t. The vertex . A . Given two elements x and y in A, x can be related to y, y can be related to x, both x related to y and y related to x can occur, or there may be no relation between x and y. 1 The digraph of a relation If A is a finite set and R a relation on A, we can also represent R pictorially as follows: Draw a small circle for each element of A and label the circle with the corresponding element of A. Representing Relations Using Digraphs • Definition: A directed graph, or digraph, consists of a set V of vertices (or nodes) together with a set E of ordered pairs of elements of V called edges (or arcs). digraph, consists of a set V of . Representing Relations using Digraphs Definition 1 A directed graph, or digraph, consists of a set V of vertices (or nodes) together with a set E of ordered pairs of elements of V called edges (or arcs). De nition 5. Representing Relations with Digraphs (directed graphs) Let R = {(a,b), (b,a), (b,c)} over A={a,b,c} We can represent R with this graph: R: a b c . M 1 ^M 2, is the zero-one matrix for R 1 \R 2. 6.3. CS340-Discrete Structures Section 4.1 Page 3 Properties of Binary Relations: R is reflexive x R x for all x∈A Every element is related to itself. De nition 6. Representing Relations using Digraphs 3. The vertex a is called the initial vertex of the edge (a;b), and the vertex b … Representing Relations Using Digraphs De nition: A directed graph, or digraph, consists of a set V of vertices (or nodes) together with a set E of ordered pairs of elements of V called edges (or arcs). Previously, we have already discussed Relations and their basic types. A directed graph, or digraph, consists of a set V of vertices (or nodes) together with a set E of ordered pairs of elements of V called directed graph, or . These circles are called the vertices. We can draw a diagram with directed line segments joining dots called a digraph to represent a relation or we can use such a directed graph to define a relation. DEFINITION 1 . Representing relations using digraphs. ICS 241: Discrete Mathematics II (Spring 2015) Meet If M 1 is the zero-one matrix for R 1 and M 2 is the zero-one matrix for R 2 then the meet of M 1 and M 2, i.e. Representing Relations Using Digraphs.