IV$.($@$3$ pts$,27+3$ total points$)$ The following questions refer to a capital budgeting problem of six projects represented by $0-1$ variables X$1,$ X$2,$ X$3,$ X$4,$ X$5,$ and X$6,$ such that$$

Xi$=1$ if project i is selected,$$

Xi$=0$ if project i is not selected.$$

i $=1,2,3,4,5,6$

Write a constraint $($or constraints$)$ to represent each restriction below $($assuming that each situation is not related to the others$).$ For each item, your constraint$($s$)$ should reflect the restriction exactly, no more no less.

$(4-$a$)$ It is not allowed to select all of the six projects.

$(4-$b$)$ At least two of the projects $2,3,4$ and $5$ must be selected.

$(4-$c$)$ Project $1$ and project $6$ must be selected or deselected simultaneously.

$(4-$d$)$ Between project $2$ and project $5,$ one must be selected, but not both.

$(4-$e$)$ At most two projects can be selected among project $1,3$ and $4.$

$(4-$f$)$ If $[$project $5$ is selected$],$ then $[$project $3$ must be selected$].$

$(4-$g$)$ If $[$Project $6$ is not selected$]$ then $[$Project $4$ must be selected$].$

$(4-$h$)$ If $[$Project $3$ is selected$]$ then $[$both Project $1$ and Project $4$ must be selected$].$

$(4-$i$)($Bonus $3$ pts$)$ If $[$Projects $1,3$ and $4$ are all selected$],$ then $[$project $5$ must be selected$].$

$(4-$j$)$ If $\{[$Project $2$ is selected$]$ or $[$Project $4$ is not selected$]\}$ then $\{$project $6$ must be selected$\}.$