$4\mathrm{.}11.$ Two$-$sender two$-$receiver channel. Consider the DM $2-$sender $2-$receiver channel depicted in Figure $4\mathrm{.}11$ with message pair $($M$\_(1),$M$\_(2))$ uniformly distributed over $[1$:$2^($nR$\_(1))]\backslash$times $[1$:$2^($nR$\_(2))].$ Sender j$=1,2$ encodes the message M$\_($j$).$ Each receiver wishes to recover both messages $($how is this channel related to the multiple access channel?$).$ The average probability of error is defined as P$\_($e$)^(($n$))=$P$(($hat$($M$)\_(11),$hat$($M$)\_(21))!=)$: $($M$\_(1),$M$\_(2))$ or :$($hat$($M$)\_(12),$hat$($M$)\_(22))!=($M$\_(1),$M$\_(2)).$ Achievability and the capacity region are defined as for the DM$-$MAC. Find a single$-$letter characterization of the capacity region of this $2-$sender $2-$receiver channel. Figure $4\mathrm{.}11.$ Two$-$sender two$-$receiver channel.