Problem $5$ We consider a set $\backslash$Omega of objects, i$.$ e$.,$ points of R$^($n$),$ that are in two different classes or subsets $\backslash$Omega $\_(1)$ and $\backslash$Omega $\_(2).$ We assume that $\backslash$Omega $\_(1)$ and $\backslash$Omega $\_(2)$ are separated by a hyperplane: there exist $\backslash$alpha inR$^($n$)$ and $\backslash$beta inR such that xin$\backslash$Omega $\_(1)$Longrightarrow$\backslash$alpha $*$x$+\backslash$beta $<0,$xin$\backslash$Omega $\_(2)$Longrightarrow$\backslash$alpha $*$x$+\backslash$beta $>0.$ a$)$ Construct an ANN with input of dimension n and output of dimension $2$ such that the network function f$($x$)=($f$\_(1)($x$),$f$\_(2)($x$))$ classifies $\backslash$Omega $\_(1)$ and $\backslash$Omega $\_(2)$ in the following sense: $\backslash$Omega $\_($i$),$i$=1,$dots,$4\backslash$alpha $,$tilde$(\backslash$alpha $)$inR$^($n$)\backslash$beta $,$tilde$(\backslash$beta $)$inR$\backslash$alpha tilde$(\backslash$alpha $)$xin$\backslash$Omega $\_(1)$Longrightarrow$\backslash$alpha $*$x$+\backslash$beta $<0$ and tilde$(\backslash$alpha $)*$x$+$tilde$(\backslash$beta $)<0,$ xin$\backslash$Omega $\_(2)$Longrightarrow$\backslash$al