I don't know if this is the full proof for the following question (1). The images are just the theorems....

I am struggling to complete it starting how I did:

This is my written proof:

Suppose we have [(x,y)] = [(0,1)].

Then (x,y) ~ (0,1)

Which implies x1=y1=0 It follows that x=0

Conversely if x-0

then [(0,y)]=[(0,1)]={0}.

Exercise 1.5.3. [Used in Theorem 1.5.5 and Theorem 15.6.] Letre Z and y e Z". (1) Prove that (x, y)] = 0 if and only if r= 0. (2) Prove that (x, y)] = 1 if and only ifx = y. (3) Prove that 0 sholds (Trichotomy Law). II. Ifr 0, then ri