Discrete Mathematics | Representing Relations

13 Dec, 2019
Prerequisite – Introduction and types of RelationsRelations are represented using ordered pairs, matrix and digraphs:
Ordered Pairs –In this set of ordered pairs of x and y are used to represent relation. In this corresponding values of x and y are represented using parenthesis.Example: {(1, 1), (2, 4), (3, 9), (4, 16), (5, 25)}
This represent square of a number which means if x=1 then y = x*x = 1 and so on.
Representing using Matrix –In this zero-one is used to represent the relationship that exists between two sets. In this if a element is present then it is represented by 1 else it is represented by 0. In this method it is easy to judge if a relation is reflexive, symmetric or transitive just by looking at the matrix.Suppose R is a relation from X={x1, x2, .....xn} to Y={y1, y2....yn}
It is represented by :-
M[i, j]={1, if (Xi, Yj) belongs to R
        0, if (Xi, Yj) does not belong to R}
If A={1, 2, 3} and B={1, 2} and Relation R isR = {(2, 1), (3, 1), (3, 2)}then all corresponding value of Relation will be represented by “1” else “0”.It is represented as:It’s corresponding possible relations are:Digraph –A digraph is known was directed graph. It consists of set ‘V’ of vertices and with the edges ‘E’. Here E is represented by ordered pair of Vertices.In the edge (a, b), a is the initial vertex and b is the final vertex.If edge is (a, a) then this is regarded as loop.Example: Suppose we have relation formingR = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} This relation is represented using digraph as:My Personal Notes
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Ordered Pairs –In this set of ordered pairs of x and y are used to represent relation. In this corresponding values of x and y are represented using parenthesis.Example: {(1, 1), (2, 4), (3, 9), (4, 16), (5, 25)}
This represent square of a number which means if x=1 then y = x*x = 1 and so on.

Example: {(1, 1), (2, 4), (3, 9), (4, 16), (5, 25)}
This represent square of a number which means if x=1 then y = x*x = 1 and so on.

Representing using Matrix –In this zero-one is used to represent the relationship that exists between two sets. In this if a element is present then it is represented by 1 else it is represented by 0. In this method it is easy to judge if a relation is reflexive, symmetric or transitive just by looking at the matrix.Suppose R is a relation from X={x1, x2, .....xn} to Y={y1, y2....yn}
It is represented by :-
M[i, j]={1, if (Xi, Yj) belongs to R
        0, if (Xi, Yj) does not belong to R}
If A={1, 2, 3} and B={1, 2} and Relation R isR = {(2, 1), (3, 1), (3, 2)}then all corresponding value of Relation will be represented by “1” else “0”.It is represented as:It’s corresponding possible relations are:
Suppose R is a relation from X={x1, x2, .....xn} to Y={y1, y2....yn}
It is represented by :-
M[i, j]={1, if (Xi, Yj) belongs to R
        0, if (Xi, Yj) does not belong to R}

If A={1, 2, 3} and B={1, 2} and Relation R isR = {(2, 1), (3, 1), (3, 2)}then all corresponding value of Relation will be represented by “1” else “0”.
It is represented as:
It’s corresponding possible relations are:
Digraph –A digraph is known was directed graph. It consists of set ‘V’ of vertices and with the edges ‘E’. Here E is represented by ordered pair of Vertices.In the edge (a, b), a is the initial vertex and b is the final vertex.If edge is (a, a) then this is regarded as loop.Example: Suppose we have relation formingR = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} This relation is represented using digraph as:My Personal Notes
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Example: Suppose we have relation forming
R = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} 
This relation is represented using digraph as:
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