I'm unsure of the definition below :
$\mathbb{Z} = (\dots,-2,-1,0,1,2,\dots)$
Take a relation $\rightarrow$ on $\mathbb{Z}^2$ define as $(x,y)$ $\rightarrow$ $(x',y')$ only when:
$(x',y')= \left(\left\lfloor \frac { |x| }{ 2 } \right\rfloor , \left\lfloor \frac { |y| }{ 2 } \right\rfloor \right)$
How do I make sense of this? Does $(x,y) \rightarrow (x',y')$ is seen like a relation between $a$ and $b$ like $a < b$?
How would you represent couples of $-5 \le (a,b) \le 5$? Considering the $\mathbb{Z}^2$.
A relation $R$ on a set $A$ is a subset of $A\times A$, and in this case we have $A=\mathbb{Z}^2$ and so $R\subset\mathbb{Z}^2\times\mathbb{Z}^2$.
Now we are told that a pair $\left((x,y),(x',y')\right)\in R$ iff the equations $x'=\left\lfloor\frac{|x|}{2}\right\rfloor$ and $y'=\left\lfloor\frac{|y|}{2}\right\rfloor$ are satisfied. So for example we know the pair $((1,-2),(0,1))\in R$.
The key point is that each element of a pair in this relation is itself a point in $\mathbb{Z}^2$.