Which are the equivalence classes for the following relation?

Question

Here I have such an exercises related to equivalence relations.

Given R defined on $Z \times Z$, $$(a,b)R(c,d)$$

and

$$a+d=b+c$$

Let set $A$ be: $$A=\lbrace{0,1,2} \rbrace$$ Which are the equivalence classes determined by the restriction of relation $R$ on set $A \times A $?

---

What I've done is: $$a-b=c-d$$

So the equivalence classes are: $$[(0,0)]=\lbrace{(0,0),(1,1),(2,2)}\rbrace$$ $$[(1,0)]=\lbrace{(1,0),(2,1)}\rbrace$$ $$[(2,0)]=\lbrace{(2,0)}\rbrace$$ $$[(0,1)]=\lbrace{(0,1),(1,2)}\rbrace$$ $$[(0,2)]=\lbrace{(0,2)}\rbrace$$

---

Can you , please, tell me if it's right what I did ? Thank you.

Answer 1

You can check for yourself.

• $\boxed{\color{green}{\checkmark}}$ Do each of the nine pairs occur in one and only one equivalence class?
• $\boxed{\color{green}{\checkmark}}$ Are all the pairs in the same class equivalent?
• $\boxed{\color{green}{\checkmark}}$ Are none of the pairs in different classes equivalent?

It looks like you have all boxes ticked to me.