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Representations of Matrices and Graphs in Relations | Engineering Mathematics - Civil Engineering (CE) PDF Download

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. This is represented as RoS.

Inverse Relation

A relation R is defined as (a, b) Є R from set A to set B, then the inverse relation is defined as (b, a) Є R from set B to set A. Inverse Relation is represented as R-1

R-1 = {(b, a) | (a, b) Є R}.

Complementary Relation

Let R be a relation from set A to B, then the complementary Relation is defined as- {(a, b)} where (a, b) is not Є R.

Representation of Relations

Relations can be represented as- Matrices and Directed graphs.

Relation as Matrices:

A relation R is defined as from set A to set B, then the matrix representation of relation is M= [mij] where

mi= { 1, if (a, b) Є R

0, if (a, b) Є R }

Properties
  1. A relation R is reflexive if the matrix diagonal elements are 1.
    Representations of Matrices and Graphs in Relations | Engineering Mathematics - Civil Engineering (CE)
  2. A relation R is irreflexive if the matrix diagonal elements are 0.
  3. A relation R is symmetric if the transpose of relation matrix is equal to its original relation matrix. i.e. MR = (MR)T.
    Representations of Matrices and Graphs in Relations | Engineering Mathematics - Civil Engineering (CE)
  4. A relation R is antisymmetric if either mij = 0 or mji = 0 when i≠j.
  5. A relation follows join property i.e. the join of matrix M1 and M2 is M1 V M2 which is represented as R1 U R2 in terms of relation.
  6. A relation follows meet property i.r. the meet of matrix M1 and M2 is M1 ^ M2 which is represented as R1 Λ R2 in terms of relation.

Relations as Directed graphs

A directed graph consists of nodes or vertices connected by directed edges or arcs. Let R is relation from set A to set B defined as (a,b) Є R, then in directed graph-it is represented as edge(an arrow from a to b) between (a,b).

Properties
  1. A relation R is reflexive if there is loop at every node of directed graph.
  2. A relation R is irreflexive if there is no loop at any node of directed graphs.
  3. A relation R is symmetric if for every edge between distinct nodes, an edge is always present in opposite direction.
  4. A relation R is asymmetric if there are never two edges in opposite direction between distinct nodes.
  5. A relation R is transitive if there is an edge from a to b and b to c, then there is always an edge from a to c.

Example:

The directed graph of relation R = {(a, a), (a, b), (b, b), (b, c), (c, c), (c, b), (c, a)} is represented as

Representations of Matrices and Graphs in Relations | Engineering Mathematics - Civil Engineering (CE)

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. R is not transitive as there is an edge from a to b and b to c but no edge from a to c.

The document Representations of Matrices and Graphs in Relations | Engineering Mathematics - Civil Engineering (CE) is a part of the Civil Engineering (CE) Course Engineering Mathematics.
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FAQs on Representations of Matrices and Graphs in Relations - Engineering Mathematics - Civil Engineering (CE)

1. What is a matrix representation of a relation?
Ans. A matrix representation of a relation is a way to represent the relationship between elements of two sets using a matrix. In this representation, the rows of the matrix correspond to the elements of one set, and the columns correspond to the elements of the other set. The entries of the matrix indicate whether there is a relationship or connection between the corresponding elements.
2. How do you represent a directed graph as an adjacency matrix?
Ans. To represent a directed graph as an adjacency matrix, we create a matrix where the rows and columns represent the vertices of the graph. If there is an edge from vertex i to vertex j, then the entry in the ith row and jth column of the matrix is 1. If there is no edge between the vertices, the entry is 0. This matrix is called the adjacency matrix of the directed graph.
3. Can a matrix representation of a relation be used to find the transitive closure of a directed graph?
Ans. Yes, a matrix representation of a relation can be used to find the transitive closure of a directed graph. The transitive closure of a graph is a matrix that represents all possible paths between vertices in the graph. By raising the matrix representation of the relation to higher powers, we can determine if there is a path of any length between two vertices. If there is a path, the corresponding entry in the transitive closure matrix will be non-zero.
4. How can matrix representations of relations be used in data analysis and machine learning?
Ans. Matrix representations of relations can be used in data analysis and machine learning to model relationships between variables or entities. For example, in social network analysis, a matrix representation of the relationships between individuals can be used to analyze patterns of connections and identify influential individuals. In machine learning, matrix representations can be used for tasks such as recommendation systems, where the matrix represents user-item interactions. Various matrix factorization techniques can then be applied to extract meaningful patterns and make predictions.
5. What are the advantages of using a matrix representation for graphs and relations?
Ans. Using a matrix representation for graphs and relations has several advantages. Firstly, it allows for efficient storage and manipulation of large amounts of data. Matrix operations can be performed quickly and efficiently using mathematical libraries. Secondly, matrix representations provide a visual and intuitive way to understand the relationships between elements. Patterns and structures in the matrix can be easily identified and analyzed. Finally, matrix representations enable the application of various mathematical techniques and algorithms, such as matrix factorization and graph algorithms, for further analysis and problem-solving.
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