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). Loops are fundamentally dull, so for the most part, we ignore them. A . Representing Relations Using Digraphs. nodes) together with a set E of ordered pairs of elements of V called . 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. 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). An edge of the form (a,a) is called a loop. Previously, we have already discussed Relations and their basic types. 6.3. 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. arcs). The vertex . Representing Relations Using Digraphs ... •Relations are sets, and therefore, we can apply the usual set operations to them. • The vertex a is called the initial vertex of the edge (a, b), and the vertex b … … 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 a. is called the . Representing relations using digraphs. 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. initial vertex . 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. of the edge (a,b), and the vertex . 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. De nition 5. •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 The vertex a is called the initial vertex of the edge (a;b), and the vertex b … now, either that is an arc from v to w in the digraph, or there isn’t. DEFINITION 1 . digraph, consists of a set V of . vertices (or . Representing Relations using Digraphs 3. directed graph, or . 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. M 1 ^M 2, is the zero-one matrix for R 1 \R 2. De nition 6. These circles are called the vertices. 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. A digraph D is a loop-digraph … edges (or . 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 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 . 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).