plz explain the meaning(definition) of these types of relations in a formal way. i could not understand the book definition.so plz explain in ur own words(an example will be appreciated)

1)transitive relation

2)symmetric and anit-symmetric relation

3)reflexive relation

thnx

rates assured :)

Reflexive relation:If a relationR is defined on the set A,then the relation is said to be reflexive iff (a,a)R for all aA.

Example:If we define a relation on A={1,2,3} then the relation R={(1,1),(2,2),(3,3),(1,3)} is a reflexive relation since (1,1),(2,2),(3,3) belong to R but R'={(1,1),(1,2),(2,2),(2,3)} is not reflexive bcoz (3,3) is not an element in it.

Symmetric relation:A relation R,defined on the set A,is symmetric iff (a,b)R(b,a) shud belong to R for a,bA.

ex:A={1,2,3} and R={(1,1),(1,2),(2,1),(2,2)} is a symmetric relation while R'={(1,2),(2,2),(3,1),(1,3)} is not a symmetric relation bcoz (1,2) is there but (2,1) does not belong to R'.

Transitive relation:A relationR,defined on A,is said to be transitive iff (a,b)R and (b,c)R(a,c)R for a,b,cA.Finally,A relation which is reflexive,transitive and symmetric is called an equivalence relation.

Ex:A={1,2,3} and R={(1,2),(2,3),(1,3)} is a transitive relation while R'={(1,2),(2,3),(2,2),(1,2)} is not transitive since (1,3) does not belong to R'

Anti-symmetric relation:A relation R,on A,is Anti-symmetric iff (a,b)R and (b,a)Ra=b.

Ex:A={1,2,3} and R={(1,2),(2,3),(1,1)} is an anti-symmetric relation while R'={(1,2),(2,1),(2,2)} is not anti-symmetric bcoz both (1,2) and (2,1) belong to R and 12.

An important thing to note is anti-symmetric does not mean not symmetric,For example,R={(1,1),(2,2),(3,3)} is both symmetric and anti-symmetric.