Different types of Relations

                Let A be a set. Then the relation R on A is defined as subset of A*A.

for example A={1,2,3}

cross product A*A={(1,1),(1,2),(1,3).(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)}

 R={(1,1),(2,3),{3,0)} is a subset if A*A. So R is relation on A

Reflexive Relation:

A relation R on A is of the form {(a,a)/a belongs to A} is called reflexive relation.

A={1,2,3}

R={(1,1),(2,2),(3,3)} is reflexive relation.

R={(1,1),(2,2)} is not reflexive since (3,3) ordered pair is missing in relation R

Symmetric Relation:
                               A relation R on A is defined as if there exist (a,b) belongs to R then (b,a) belongs to R such a relation is called symmetric  relation

A={1,2,3}

R={(1,1),(1,2),(2,1),(3,1),(1,3)} is a symmetric relation

R1={(1,1),(1,2),(2,1),(3,1),(3,3)} is not a symmetric relation since (3,1) belongs to R1 but (1,3) ordered pair is missing in relation R


Transitive Relation:

    The transitive relation is defined as if (a,b)belongs to R and (b,c) belongs to R then (a,c) belongs to R

Let A={1,2,3}

R={(1,1),(1,2),(2,1),(3,1),(1,2),(3,2)} is a transitive relation.

R1={(1,1),(1,2),(2,1),(3,2)} is not transitive since (3,2) and (2,1) belongs to R but there is not ordered pair (3,1) in R1

Anti Symmetric Relation:
  A relation R on A is defined as if there exist (a,b) belongs to R then (b,a) does not belongs to R such a relation is called anti symmetric relation

A={1,2,3}

R={(1,1),(1,2),(2,1),(3,1),(1,3)} is not anti symmetric relation because for example (1,2) and (2,1) belongs to R

R1={(1,1),(1,2),,(3,1),(3,3)} is anti symmetric relation,


Note: An identity relation is reflexive, symmetric, anti symmetric and transitive

Equivalence relation:

                      A relation is said to be transitive relation if it is reflexive, symmetric and transitive.

R={(1,1),(1,2),(2,1)} is transitive relation

Partial Ordered Set(POSET):
                   A relation is said to partial ordered set if it is reflexive, anti symmetric and transitive relation .

R1={(1,1),(1,2),(2,2),(3,2)} is partial ordered set .



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