Writing an equivalence relation defined on $ℚ$.

Question

I need to write a equivalence relation defined on $ℚ$ to partition it on two equivalent classes. I think the best way is to split it on negative and non-negative numbers.

$A = \{ \frac{m}{n}∣ m,n ∈ ℤ, m ≥ 0 > n\}$

$B = \{\frac{m}{n}| m,n ∈ ℤ, m,n>0\}$

There will be all negatives numbers in $A ∪ \{0\}$ and all non-negative numbers in $B$.

$R = \{ (a,b) |$ where $a,b$ are both positive or negative $\}$

That my idea how this relation should look like. Is it right?

Answer 1

The way you have set things up is so that all of the non-positive numbers will be in $A$, and all of the positive numbers will be in $B$. In particular, $0$ is not negative, so it can't be the case that all and only the negative numbers are in $A$.

Remember, $R$ is an equivalence relation if it satisfies:

1. $$xRx$$
2. $$xRy ⟺ yRx$$
3. $$xRy ∧ yRz ⟹ xRz$$

It seems like what you are looking for is to interpret $R$ as "has the same sign (negative or non-negative) as", in which case yes $R$ is an equivalence relation on $ℚ$. But you need to be careful about your wording, because the way you wrote $R$ does not consider $0$.

Let me know if I can help you any further. Mshvidobit :-)