$4\mathrm{.}11.$ 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

$[1$:$2^{n{R}_1}][1$:$2^{n{R}_2}].$ 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 $P_e^{(n)}=P\{\begin{array}{ccc}(hat(M{)}_{11},& \text{h}\text{a}\text{t}& (M{)}_{21})\end{array}$

$(M_1,M_2)$ 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.Please provide solution for this question.