1Give an example to show that when the symmetric closure of

1)Give an example to show that when the symmetric closure of the reflexive closure of the transitive closure of a relation is formed, the result is not necessarily an equivalence relation.

and

2) Let R be the relation on Z+ defined by xRy if and only if x < y. Then, in the Set Builder Notation, R = {(x, y) | y x > 0}.

Use the Set Builder Notation to express the composite relation R^n , where n is a positive integer.

Solution

1) Let R={ (1,2), (1,3)}, then transitive closure of R is R⁺={ (1,2),(1,3)} and

reflexive closure of R⁺is R⁺⁺={(1,2),(1,3),(1,1),(2,2),(3,3)} and

symmetry closure of R⁺⁺ is R⁺⁺⁺ ={ (1,2) (1,3) (1,1) (2,2) (3,3) (2,1) (3,1)}

but R⁺⁺⁺is not a equivalence relation because it is not satifies transitive.( since (3,1) and (1,2) arein R⁺⁺⁺but (3,2) is not in that relation.)

2) Let R={(x,y)| y-x>o} then Rⁿ={(x,y)| y-x>n}.