Question

Consider the equivalence relation R = {( x, y) Ix-y is an integer}.

(a) What is the equivalence class of 1 for this equivalence relation?
(b) What is the equivalence class of 1/2 for this equivalence relation?

Answer 1

[1]=Z the set of integers

[1/2]= r is an odd integer

Explanation:

Denote by [a] the equivalence class of an element a.

We know that [a]=(x,a)∈R. Then

[1]=(x,1)∈R=x=x

=x=k∈Z={...,-2+1,-1+1,0+1,1+1,2+1,...}=Z

For the other class, we have

[1/2]=(x,1/2)∈R=x-1/2 is an integer=x-1/2=r for some r∈Z

=x=r∈Z={...,-2+1/2,-1+1/2,0+1/2,1+1/2,..}

={...,-3/2,-1/2,1/2,3/2,...}=r/2