Unicourse is arranging a school trip for a class consists of $20$ students. However, there are some restrictions that has to be satisfied. To fulfill these constraints, the university asks you to formulate a set of linear constraints. Think about each of these restrictions independently.

i$.$ At least $12$ students should join that school trip among $20$ students.

ii$.$ If student number $1$ is coming, then student number $4$ should also come.

iii. If student number $2$ is coming, then student number $3$ should not come.

iv$.$ If student number $6$ is coming, then either student number $11$ or student number $12$ should come but not both.

v$.$ Either both students $5$ and $7$ comes or neither of them.

vi$.$ Unless student $2$ comes, student $3,$ student $4,$ and student $5$ cannot come. If student $2$ comes, then at most two and at least $1$ of student $3,$ student $4$ and student $5$ should come.

vii. At least one of the students $\backslash (\backslash \{18,19,20\backslash \}\backslash )$ or at least two of the students $\backslash (\backslash \{9,10,11,12\backslash \}\backslash )$ should come.

viii. If at least two of the students $\backslash (\backslash \{1,3,5,7\backslash \}\backslash )$ are coming then at most one from the students $\backslash (\backslash \{2,4,6\backslash \}\backslash )$ should come.