Check the following Relation $R=\{(x,y) |\exists k\in \mathbb{Z} \cdot x*y=3k \}$

Question

I would like to check the following relation: $$R=\{(x,y) |\exists k\in \mathbb{Z} \cdot x*y=3k \},R\subseteq \mathbb{Z} \times \mathbb{Z}$$

1. Reflexivity
2. Symmetric
3. Transitivity
4. Asymmetric
   

Can I write a matrix for this relation? its possible?
What is an element in this relation? $(x,y)$ is $1$ element? for example if I want to check relexivity what should I take? $(x,y)R(x,y)$? OR $(x,x)R(x,x)$?
Suggestions are welcomed! Thanks.

Answer 1

We have $\left(1,1\right)\notin R$. This shows that $R$ not reflexive.

We have $xy=yx$ so if $xy=3k$ then also $yx=3k$. This shows that $R$ is symmetric.

We have $\left(1,3\right)\in R$ and $\left(3,1\right)\in R$ but not $\left(1,1\right)\in R$. This shows that $R$ is not transitive.

We have $\left(1,3\right)\in R$ but not $\left(3,1\right)\notin R$. This shows that $R$ is not asymmetric.

I don't know what a matrix for a relation is, so cannot help you here.

Answer 2

Assuming $x,y \in \mathbb{Z}$.

1. It is not reflexive. $$ (1,1) \notin R \because 1*1 = 1 \neq 3k \ (k \in \mathbb{Z}) $$
2. It is symmetric. $$ (x,y) \in R \implies x*y = 3k \ (k \in \mathbb{Z}) \implies y*x = 3k \implies (y,x) \in R. $$
3. It is not transitive. $$ (1,3) \in R, (3,7) \in R \ \ \text{but} \ (1,7) \notin R $$
4. It is not asymmetric. $$ (1,3), (3,1) \in R $$

Answer 3

The relation is a relation on $\Bbb Z$, this means that we want to ask whether or not $3\mathrel{R}3$, rather than asking $(3,3)\mathrel{R}(3,3)$.

As for a matrix, you probably can't, on the count that this is an infinite matrix, and you won't have time to finish it, or matter to write on (and with) the entire matrix. But it is probably worth a shot writing it for a few elements (look at the relation restricted to four-five elements).

Doing so is likely to give you the counterexample needed for reflexivity, asymmetry and transitivity, and might give you a hunch as to why this is a transitive relation.