Let $A,B$ be sets. Determine whether the following statement is true: If $f$:$xY$ and $g$:$YZ$ are functions and $g$@$f$ is one$-$to$-$one, must $g$ be one$-$to$-$

one? Justify your answer. Determine whether or not the following is true:

$(x-y)|(x^n-y^n),AAnin{Z}^{+}$and $x,$yinZ,$xy$; Let $A,B$ and $C$ be three $(n,n)-$matrices. Determine whether the following is true:

$A*(B+C)=A*B+A*C.($a$)$ Let $A=([0,1,0],[1,1,2],[0,2,0])$ find $A^2$ and draw the digraph corresponding to $A^2.$

$($b$)$ Find the adjacency matrix for the following directed graph:

$($a$),(\frac{A^c{B}^c}{\frac{?}{?}}A{)}^c=A$

$($b$),(\frac{A}{\frac{?}{?}}C)(\frac{B}{\frac{?}{?}}C)(\frac{A}{\frac{?}{?}}B)=\frac{O}{?}$;