(Solved): Consider the binary relations N4 and N8 on the set of all pixels defined by: (x,y)N4(x2,y2 ...

Consider the binary relations ${\mathcal{N}}_4$ and ${\mathcal{N}}_8$ on the set of all pixels defined by: $\begin{array}{l}(\mathrm{x},\mathrm{y}){\mathcal{N}}_4\left({\mathrm{x}}^2,{\mathrm{y}}^2\right)??\mathrm{x}?{\mathrm{x}}^3?+?\mathrm{y}?{\mathrm{y}}^{?}?=1\\ (\mathrm{x},\mathrm{y}){\mathcal{N}}_8\left({\mathrm{x}}^2,{\mathrm{y}}^2\right)?\max \left\{?\mathrm{x}?{\mathrm{x}}^3?,?\mathrm{y}?{\mathrm{y}}^{?}?\right\}=1\end{array}$ If $(\mathrm{x},\mathrm{y}){\mathcal{N}}_4\left({\mathrm{x}}^{?},{\mathrm{y}}^{?}\right)$, we say that $(\mathrm{x},\mathrm{y})$ is a 4-neighbour of $\left({\mathrm{x}}^{?},{\mathrm{y}}^{?}\right)$. If $(\mathrm{x},\mathrm{y}){\mathcal{N}}_8\left({\mathrm{x}}^{?},{\mathrm{y}}^{?}\right)$, we say that $(\mathrm{x},\mathrm{y})$ is an 8 -neighbour of $\left({\mathrm{x}}^{?},{\mathrm{y}}^{?}\right)$. (a) Assume $\mathrm{M}=9$ and $\mathrm{N}=25$, as in Fig. 1. What are the 4-neighbours of $(0,0)$ ? What are the 4-neighbours of $(1,0)$ ? What are the 8-neighbours of $(1,0)$ ? What are the 8-neighbours of $(1,1)$ ? (b) Is ${\mathcal{N}}_4$ reflexive, symmetric, antisymmetric, transitive? ${\mathcal{N}}_8$ has the same properties as ${\mathcal{N}}_4$, but you are not asked to prove it.