$($a$)$ Let $f$ and $g$ be the functions defined from the set of integers into the set of integers by $f(x)=11x+6$ and $g(x)=5+7x.$ Determine the compositions $(f$@$2g)(x)$ and $(g$@$f)(3x)?$ Find a value of $x($if any$)$ such that

$(g$@$f)(3x)=(f$@$2g)(x)+8.$

$(2$ marks$)$

$($b$)$ Let $A=\{3,5,9,10,15,27,30,45\}.$ Draw the Hasse diagram for the partial order "divisibility $|"$ on A$.$ Find maximal and minimal elements of the poset $(A,1).$

$(2$ marks$)$

$($c$)$ Using Warshall's algorithm, find the transitive closure of the relation $R=\{(2,3),(3,4),(3,6),(4,5),(5,3)\}$ on $A=\{2,3,4,5,6\}.$

$(2$ marks$)$