1. ## Relation Question

Let $R$ be a relation on $\mathbb{Z}$, where $xRy \Leftrightarrow x|9y$. State whether R is reflexive, symmetric, transitive or an equivalence relation.

I'm having trouble understand the notation | in the question. I am used to this meaning "such that". The text doesn't explain this problem very well; it says that x must = 9ky were $k \in \mathbb{Z}$

How do I go about determining the relation here?

Thanks.

2. ## Re: Relation Question

Originally Posted by terrorsquid
Let $R$ be a relation on $\mathbb{Z}$, where $xRy \Leftrightarrow x|9y$. State whether R is reflexive, symmetric, transitive or an equivalence relation.
I'm having trouble understand the notation | in the question. I am used to this meaning "such that". The text doesn't explain this problem very well; it says that x must = 9ky were $k \in \mathbb{Z}$
The notation $a|b$ usually means that $a\text{ divides }b$.
But that does really make sense here. The textbook should have defined it somewhere in the text.

If it does mean divides the the relation cannot be reflexive because 0 does not divide 0.

3. ## Re: Relation Question

It probably means "divides." It's not symmetric since 3R2, but not 2R3.

4. ## Re: Relation Question

R is reflexive since $x|9x$

R is not symmetric since if x = 2 and y = 18, then $2|162$ but 2 is not divisible by 18.

R is not transitive since if x = 9y and y = 9z then x = 81z, which clearly does no divide 9z. (I don't understand why 81z doesn't divide 9z? is it something to do with the fact that R is is on $\mathbb{Z}$ ?)

5. ## Re: Relation Question

So, knowing that | means divides, what is the best method to determine the nature of the relationship? Just to take arbitrary integers and try to find a contradiction?

6. ## Re: Relation Question

Originally Posted by terrorsquid
R is reflexive since $x|9x$
The reason is that for every $x\in\mathbb{Z}$ we have $9x=9\cdot 1\cdot x$ .

7. ## Re: Relation Question

More simply for transitive: Use x=81, y=9, z=1. Then xRy (because 81|81), yRz (because 9|9), but not xRz (because 81 does not divide 9).

8. ## Re: Relation Question

Hmmm... is 0|0 true? I think the definition of "divides" is a|b if there is an integer k such that b=ka. Under this definition 0|0 is true since 0=k*0 for ANY integer k. Can someone with a better memory than me remind me if k must be unique.

9. ## Re: Relation Question

Originally Posted by DrSteve
Hmmm... is 0|0 true?
Yes , $0= 1\cdot 0$ i.e. there exists $k\in\mathbb{Z}$ such that $0=k\cdot 0$ .

10. ## Re: Relation Question

Originally Posted by terrorsquid

R is not symmetric since if x = 2 and y = 18, then $2|162$ but 2 is not divisible by 18.
With the answer given here, I understand the first test 2 divides 9*(18)... but why is the second test 18 divides 2? Shouldn't it be x = 18 and y = 2 and $18 | 9(2)$ which is true?

11. ## Re: Relation Question

Originally Posted by terrorsquid
I'm having trouble understand the notation | in the question. I am used to this meaning "such that". The text doesn't explain this problem very well; it says that x must = 9ky were $k \in \mathbb{Z}$
This problem is creating confusion. Taking into account that $a|b$ is an universal agreement for $a$ divides $b$, Did you mean $9y=kx$ where $k\in\mathbb{Z}$?. Could you transcribe the exact formulation?

12. ## Re: Relation Question

Originally Posted by FernandoRevilla
This problem is creating confusion. Taking into account that $a|b$ is an universal agreement for $a$ divides $b$, Did you mean $9y=kx$ where $k\in\mathbb{Z}$?. Could you transcribe the exact formulation?
On this webpage is the following quote:
"Clearly, $1|n$ and $n|n$. By convention, $n|0$ for every $n$ except 0 (Hardy and Wright 1979, p. 1)"

13. ## Re: Relation Question

Originally Posted by Plato
"Clearly, $1|n$ and $n|n$. By convention, $n|0$ for every $n$ except 0 (Hardy and Wright 1979, p. 1)"
Let us see. Conventions are conventions. If we define in $\mathbb{Z}$ $x|y$ iff $y=kx$ for some $k\in\mathbb{Z}$ then, $|$ is reflexive (no problem). If we exclude $0|0$ then $|$ is not reflexive by convention (no problem). That depends on the context: in the context of Measure Theory $0\cdot \infty =0$ (by convention), in the context of infinite products $1^{\infty}=1$ (by convention) etc. For those reasons and looking at all the previous answers I suggested to terrorsquid to transcribe the exact formulation of the problem, taking (besides) into account that he,or the problem (who knows?) says $x|y\Leftrightarrow x=9ky$ for some $k\in\mathbb{Z}$ .

14. ## Re: Relation Question

I wrote the question down word for word in my original post. The next part I mentioned was from the explanation of the solution - not the question itself. I wont be able to reference the exact problem again as the questions are randomly generated in our online text quiz section with unique solution explanations. I could reference another similar question from the same section; it will depend on which question the quiz generates but they are always the same type of question just with slightly different values.