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| r is an odd integer}

Step-by-step explanation:

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

We know that [a]={x|(x,a)∈R}. Then

[1]={x|(x,1)∈R}={x|x-1 is an integer}={x|x-1=k for some k∈Z}

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

For the other class, we have

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

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

={...,-3/2,-1/2,1/2,3/2,...}={r/2| r is an odd integer}

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