$1.$ Often in binary classification we are interested in the differences in the output of our current classifier g and an unknown function f that we are trying to learn. It is common in these cases to examine the quantity f$($x$)$g$($x$).$ Let D be an arbitrary distribution on the domain $\{-1,1\}",$ and let f$,$g:$\{-1,1\}"+\{-1,1\}$ be two Boolean functions. Prove that $1-$ Ex~D$[$f $($x$)$g$($x$)]$ Pe~D$[$f$($x$)+$ g$($x$)]2$ Would this still be true if the domain were some other domain $($such as R$",$ where R denotes the real numbers, with say the Gaussian distribution$)$ instead of $\{-1,1\}"?$ If yes, justify your answer. If not, give a counterexample. Note only the domain is changing; the output is still Boolean.