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?

[1]=Z the set of integers

[1/2]=r/2

Step-by-step explanation:

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

We know that [a]=x. Then

[1]=x=x=x-1=k for some k∈Z

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

For the other class, we have

[1/2] = (x, 1/2) ∈R=x=x

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

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