Realtions

              

  In mathematics relation has same meaning as in our daily life we associate someone with someone else by certain rule.

Considering the following Example:

                                                                                                       

                       In the above example A is related to Adam D is related to Davood and I is related to Ibrahim, because in set y every name

or every subset starts with letter I,A and D respectively. These alphabets are present in set X. Hence X is related to Y and Y is relate to X.

How to Define of Relation: Let A&B be any two non empty sets, then any subset of AxB is called a relation from A to B.

Important terms to understand before study of relation:

1)      Domain: If A&B are two sets then all elements present in A which are related to B are called domain of R. In other words first

entries of ordered pair are called domain of R

2)      Range: If A&B are two non empty sets then elements of B which are related to A are called range.

3)      Co Domain: If A&B are two non empty sets then set B is called co domain of R.

4)      Inverse relation: If R be a relation from A to B then teh inverse relation of R denoted as R^-1 is a relation from B to A defined by

      R^-1 = {(y,x):XEA & YEB, (X,Y) E R}

Thus inverse of relation consists of those ordered pairs which when reversed belong to R i.e., Y R^-1 X iff X R Y

Reflexive:  If A&B are two sets then Relation “R” from A to  B is said to be reflexive iff,

(a,a) E R i.e., A has some image in B

Symmetric: If A&B are two sets then Relation ‘R’ from A to B is said to be symmetric isff,

Inverse of relation exists i.e., if (a,b) E R then (b,a) also E R i.e., R=1/R or R=R^-1

Transitive: A relation is said to be transitive iff,

(a,b) E R    &     (b,c) E R

= (a,c) E R

Anti Symmetric: A relation is said to be anti Symetric when

(a,b) E R     &     (b,a) E R

= a=b