Let R be a relation on R given by xRy if cosx cosy a Prove

Let R be a relation on R, given by xRy if cosx = cosy.

(a) Prove that R is an equivalence relation.

(b) Find the equivalence classes [0] and [1].

Solution

(a) Prove that R is an equivalence relation.

let x belong to R

reflexive cos x = cos x

hene reflexive

symmetric

the relation can be written as

cosx -cosy =0

now if cosx -cosy =0 then cosy -cosx =0

hence symmetric

transitive

if cosx -cosy =0 and cosy - cosz =0

then cos x = cos z

hence transitive

therefore equivalence relation

b)Find the equivalence classes [0] and [1].

[0] when x=y = npi/2where n =2n+1 , n= 0, 1, ..........

[1] when x =npi where n = 2n+1