plz tell me what is equivalence relation? with the help of solved example?

Dear Student,

An equivalence relation on a set X is a subset of X x X , i.e., a collection R of ordered pairs of elements of X, satisfying certain properties. Write "x R y" to mean (x , y) is an element of R, and we say "x is related to y," then the properties are

1. Reflexive: a R a for all a in X,

2. Symmetric: a R b implies b R a for all a and b in X

3. Transitive: a R b and b R c imply a R c for all a,b and c in X.

Examples:

1. Two counting numbers are equivalent if they have the same parity -- that is if they are both even or both odd. Here 4 is related to 6 (since both are even) but not to 7.

    

2. Two counting numbers are equivalent if they leave the same remainder after division by 3. Here 4 is related to 7 (since both are 1 more than a multiple of 3) but not to 6.

Regards Topperlearning.