Is $R = \{ (b, c) : b, c \in \mathbb{Z}, bc > 0 \}$, a relation on $\mathbb{Z}$, an equivalence relation?

Question

I'm trying to figure out how I can describe this relationship so that I can see if it's reflexive/irreflexive/symmetric/antisymetric/transitive

If it was b - c < 0 or something I'd easily be able to understand. But I don't see what I can say definitively about b and c in this set other than that they must be either both positive or both negative in order for the condition to be true.

Answer 1

No

For our relation to be an equivalence relation we need it to be reflexive which means that $(x,x) \in R$ $\forall x \in \mathbb{Z}$.

Choose $x = 0$, then $x^2 = 0 \not > 0 \therefore (0,0)\not \in R$

Now if you meant $bc \geq 0$ instead, then we actually still do not have an equivalence relation. Choose $(-1, 0), (0, 2) \in R$. If $R$ was an equivalence relation then $(-1, 2)\in R$ via transitivity. However $-1\cdot 2 \not \geq 0$ therefore $R$ cannot be an equivalence relation.

Answer 2

A relationship needn't be neat and nice.

Here $7$ is related to $257493758943789$ simply because $7*257493758943789>0$.

And $7$ is not related to $-1$ because $-1*7 \le 0$.

And $-1$ is related to $-27$ because $-1*(-27) > 0$.

...

Is it equivalent?

Well is it reflexive.

Is $a$ always related to itelf? Is $a*a = a^2 > 0$ always.... well not if $a=0$. $0*0 = 0$ so $0*0 \not > 1$ so $0$ is not related to itself.

This is not an equivalence relation.

We don't need to but it'd be illuminating to see if it reflexive.

If $a*b > 0$ does it follow that $b*a > 0$. Well, yes.....

And transitive.

If $a*b > 0$ and $b*c > 0$ does it follow that $a*c > 0$?

Well. $a*b > 0$ and $b*c > 0$ so $a*b*b*c > 0$ and as $a*b \ne 0$ then $b\ne 0$ so $b^2 > 0$ so $\frac {a*b*b*c}{b^2} = ac > 0$.

Or could have done cases. $a*b > 0$ so either 1) $a>0$ and $b>0$ or 2) $a< 0$, $b < 0$. If 1) then $b*c > 0\implies c > 0$ and $ac > 0$. If 2) then $b*c > 0\implies c < 0$ and $ac > 0$.

....

Notice that if it weren't for the $0$ this would be an equivalence relationship.

So on $\mathbb Z\setminus\{0\}$ this is an equivalence relationship.

And what are the equivalence classes?

Well, $a$ is related to $b$ if $ab > 0$ and that happens if and only if $a$ and $b$ are both positive or both negative.

So all the positive numbers are related to each other. And all the negative numbers are related to each other.

So there are $2$ equivalence classes:

A) $\{z\in \mathbb Z| z < 0\}$

and B) $\{z\in \mathbb Z| z > 0\}$.

......

But I don't see what I can say definitively about b and c in this set other than that they must be either both positive or both negative in order for the condition to be true.

Which is enough. $a$ and $b$ are related if 1) they are both positive or both negative 2) they are both the same sign 3) the are both on the same side of zero 4) $ab > 0$ or 5) they are both either greater or both less than zero

are all perfectly good and valid relations (not equivalent if you allow $0$ but valid relations).

Why do you seem to think that's not enough.