More Crisp Relation Terms

Any relation between two sets X, Y is known as binary relation. When X ≠ Y, binary relation R(X,Y) is called bipartitie graph. Whereas if X = Y, then R(X,Y) is called directed or di-graph.
Binary relations are represented by relation matrices and also by sagittal diagrams. Here each of the sets (X,Y) are represented by a set of nodes in the diagram.
Nodes corresponding to one set are clearly distinguished from nodes corresponding to another set.

Domain :-

The domain of a crisp binary relation R(X,Y) is defined as crisp subset of X whose members participate in the relation. It is written as below:

Range :-

The range of a crisp binary relation R(X,Y) is defined as crisp subset of Y whose members participate in the relation. It is written as below:

Inverse of Crisp Relation :-

The inverse of a crisp relation R(X,Y) is a subset of cartesian product of Y, X ie, Y x X and is written and represented as

Tolerance and Equivalence Relation :-

A crisp relation R(X, X) which satisfies the properties of reflexivity and symmetry is a tolerance relation.
A crisp relation R(X, Y) which satisfies the properties of reflexivity, symmetry and transitivity is called an equivalence relation.

Reflexive :- A crisp relation R(X, Y) is reflexive, if and only if (x,x) ∈ R ∀ x ∈ X. In other words, if every element of X is related to itself then that relation is reflexive, if not it is anti-reflexive.
Symmetry :- A crisp relation R(X, Y) is symmetric if and only if for every (x,y) ∈ R then there should be (y,x) ∈ R ∀(x,y) ∈ X. And any relation that is not symmetric is called asymmetric relation.
If (x,y) ∈ R and (y,x) ∈ R ⇒ x = y
then the relation is called anti-symmetric.
If either (x,y) ∈ R or (y,x) ∈ R whenever x ≠ y then the relation is called strictly antisymmetric.
Transitive :- A crisp relation R(X, Y) is transitive if and only if (x, z) ∈ R whenever (x, y) ∈ R and (y, z) ∈ R for atleast one y ∈ Y.

Example :-

Let X be set of courses offered by an university and a relation R is defined as “prerequisite for“. Then R is

Anti-reflexive because same courses never be prerequisite for same.
It is also asymmetric because for studying higher degree lower degree is prerequisite but vice-versa is not true.
It is transitive because when we consider the example – If for studying B.Tech 10^th class is prerequisite and for studying M.Tech/M.S B.Tech is prerequisite. Then obviously for studying M.Tech 10^th class is also a prerequisite.

Operations on Crisp Relations

Let R and S be two relations defined on X x Y and are represented by relational matrices. The following operations can be performed on these relations R and S