Digraphs and Relations

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1 Digraphs and Relations RAKESH PRUDHVI KASTHURI 07CS1035 Professor: Niloy Ganguly Department of Computer Science and Engineering IIT Kharagpur November 6, 2008

2 November 6, 2008 Section 1 1 Digraphs A graph in which each graph edge is replaced by a directed graph edge, also called a digraph. A directed graph having no multiple edges or loops (corresponding to a binary adjacency matrix with 0s on the diagonal) is called a simple directed graph. A complete graph in which each edge is bidirected is called a complete directed graph. A directed graph having no symmetric pair of directed edges (i.e., no bidirected edges) is called an oriented graph. A complete oriented graph (i.e., a directed graph in which each pair of nodes is joined by a single edge having a unique direction) is called atournament. If G is an undirected connected graph, then one can always direct the circuit graph edges of G and leave the separating edges undirected so that there is a directed path from any node to another. Such a graph is said to be transitive if the adjacency relation is transitive. A digraph looks like Digraph of a Relation: If A is a finite set and R is a relation on A, then we can also represent R as in above digraph where A = 1, 2, 3, 4 R = (1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (2, 4), (3, 4), (4, 1) 2

November 6, 2008 Section 1 1.2 Example Find the relation detrmined by the fallowing digraph.

3 November 6, 2008 Section Example Find the relation detrmined by the fallowing digraph. Solution:: Since a i Ra j if and only if there is an edge from a i to a j,then we have R = (1, 1), (1, 3), (2, 3), (3, 2), (3, 4), (4, 3) simply digraphs are nothimg but the geometric representations of relations. 1.3 Example Find the matrix M R detrmined by the above digraph. Solution: put 1 at (i,j) if a i Ra j...and otherwise 0. M R =

4 November 6, 2008 Section 2 2 Types of Relation 1.Reflexive & Irreflexive relations 2.Symmetric, Assymmetric & Antiymmetric Relations 3.Transitive Relations 2.1 Reflexive and Irreflexive Relations: In set theory, a binary relation can have, among other properties, reflexivity or irreflexivity. At least in this context, (binary) relation (on X) always means a subset of XX, or in other words a function from a set X into itself. If a relation is reflexive, all elements in the set are related to themselves. For example, the relations is not greater than and is equal to are reflexive over the set of all real numbers. Since no real number is greater than itself, if you compare any number to itself, you will find is not greater than to be true. Since every real number is equal to itself, if you compare any number to itself, you will find is equal to to be true. A reflexive relation is ON set X. This means that all elements in a set are related to themselves by the relation. There are relations which are reflexive on certain sets but not reflexive on the set of real numbers. Say the relation is: a is related to b if (a - b/2) is a whole number. This relation is reflexive on the set of EVEN numbers but not reflexive on the set of real numbers. Because /2 = 2-1 = 1 is a whole number 4-4/2 = 4-2 = 2 is a whole number BUT 3-3/2 = = 1.5 is NOT a whole number. Formally: * A reflexive relation R on set X is one where for all a in X, a is R-related to itself. In mathematical notation, this is: 4

5 November 6, 2008 Section 2 a X, ara. * An irreflexive (or aliorelative) relation R is one where for all a in X, a is never R-related to itself. An irreflexive relation is a relationship for which no element of a set is related to itself. Formally: a X, (ara). Note:For a reflexive relation R on A...diagonal elements in M R are1 s. 2.2 Symmetric, Assymmetric & Antisymmetric Relations A binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a. In mathematical notation, this is: a, b X, arb bra. Asymmetric often means, simply: not symmetric. In this sense an asymmetric relation is a binary relation which is not a symmetric relation. In some texts the word is given the following stronger definition. A relation R on X is asymmetric in the following sense. * If, for all a and b in X, if a is related to b, then b is not related to a. In mathematical notation, this is: a, b X, arb (bra). Being asymmetric in this sense is the same as being both antisymmetric and irreflexive. A binary relation R on a set X is antisymmetric if, for all a and b in X, if a is R to b and b is R to a, then a = b. so, a, b X, arbbra a = b or equally, a, b X, arba b bra. Inequalities are antisymmetric, since for numbers a and b, a b and b a if and only if a = b. The same holds for subsets. 5

c) the usual digraph as shown... but in the fallowing Digraph each undirected edge corresponds two ordered pairs in relation R.

6 November 6, 2008 Section Example: Let A = 1, 2, 3, 4 and R is symmetric relation given by R = (a, b), (b, a), (c, a), (a, c), (b, c), (c, b), (b, e), (e, b), (e, d), (d, e), (c, d), (d, c) the usual digraph as shown... but in the fallowing Digraph each undirected edge corresponds two ordered pairs in relation R. 3 Transitive Relation A binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c. Transitivity is a key property of both partial order relations and equivalence relations. Mathematically a, b, c X if arb, brc then arc 6

7 November 6, 2008 Section example let A = 1, 2, 3 and let R be any Relation on A whose matrix is M R = show that R is transitive Solution: By direct computation M 2 R = M R...so, R is Transitive 7

1. Represent each of these relations on {1, 2, 3} with a matrix (with the elements of this set listed in increasing order).

Exercises Exercises 1. Represent each of these relations on {1, 2, 3} with a matrix (with the elements of this set listed in increasing order). a) {(1, 1), (1, 2), (1, 3)} b) {(1, 2), (2, 1), (2, 2), (3,