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Mathematics | Representations of Matrices and Graphs in Relations
Previously, we have already discussed Relations and their basic types.
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 MR= [mij] where
mij = { 1, if (a,b) Є R
0, if (a,b) Є R }
Properties:
Relations as Directed graphs:
A directed graph consists of nodes or vertices connected by directed edges or arcs. Let R is a relation from set A to set B defined as (a,b) Є R, then in directed graph-it is represented as an edge(an arrow from a to b) between (a,b).
Properties:
Example:
The directed graph of relation R = {(a,a),(a,b),(b,b),(b,c),(c,c),(c,b),(c,a)} is represented as :
Since there is a 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.
Representations of Matrices and Graphs in Relations
ASSIGNMENT : Engineering Mathematics III Representations of Matrices and Graphs in Relations Assignment MARKS : 10 DURATION : 1 week, 3 days