Which of the following are equivalence relations Fore the eq

Which of the following are equivalence relations? Fore the equivalence relations, identify or describe the equivalence classes. For the other relations, specify which properties, reflexivity, symmetry, and transitivity, may fail.

6. Which of the following are equivalence relations? For the equivalence relations, identify or describe the equivalence classes. For the other relations, specify which properties, reflexivity, symmetry, and transitivity, may fail. (a) For m,n EN, define (m,n) E R if m+ n is an even integer. (b) Let V be the set of vertices of a graph G, and for u,v s t V define (u, v) € R2 if“ (u, v) R2 if u- v or there exists a path from u to v. or there exis

Solution

a)the relation m+n

reflexive relation :- m+m=2m i.e, it is always even number satisifies

symmetric relation:- m+n=n+m satisifies

transitive:- m+n is even , n+o is even

=>either m,n and n,o are both odd or both even

case i)

m,o are odd

=>m+o is even

case ii)

m,o are even

=> m+o is even

it satisifies all three reflexive , symmetric and transitive relations

=> it is equivalance relation

b) reflexive :-

every vertex equal to its vertx i.e, u=u

symmetric :-

u=v => v=u

or

if path exists between u and v then there exists path between v and u

hence symmetric

transitive:-

if there is a path between u, v and v,w then automatically there exists a path between u and w

hence transitive

there fore equivalance