I am having problem understanding these concepts.
For example, let $A = \{2,3,4,5,6,7,8\}$.
The definition I found says that $x R y \iff 3 | (x-y)$.
How do I know if the relation $R$ on $A$ is reflexive, symmetric or transitive?
Thank you for your help!
This concernes the relation $R$. It is:
Relexive since: $\forall x \in A \quad x R x$ : $x-x=0=3.0$ then $3 | (x-x)$.
Symetric : If $x,y \in A$ then $3 | (x-y) \Rightarrow 3 |(y-x)$
Explanation: if $3 | (x-y)$ there existe $k \in \Bbb Z$ such that $x-y=3k$, then $y-x=3.(-k)$
Transitive: if $x,y,z \in A$ then $3 |(x-y) \, \text{and} \, 3 | (y-z) \Rightarrow 3| (x-y)+(y-z)=x-z $
Expl: $x-y= 3k, y-x=3\ell$ for $k,\ell \in \Bbb Z$ then : $x-z=(x-y)+(y-z)=3(k+\ell)$