$[8$ marks$]4.$ Prove that $(\frac{R}{\frac{?}{?}}\{0\},)$ has the following properties: $(1)$ closed; $(2)$ identity; $(3)$ inverse. Do you think

this set and operation also has the property of commutivity? Why or why not? $[$note: you do not need

to give a proof, only explain your intuition$]$

$[6$ marks$]5.$ Suppose $,,,in{S}_8$ such that

$($@${)}^{-1}$@$=$@${}^{-1}$

where $=\left[\begin{array}{cccccccc}1& 2& 3& 4& 5& 6& 7& 8\\ 2& 1& 7& 8& 3& 4& 5& 6\end{array}\right],=\left[\begin{array}{cccccccc}1& 2& 3& 4& 5& 6& 7& 8\\ 7& 3& 1& 4& 2& 5& 8& 6\end{array}\right],=\left[\begin{array}{cccccccc}1& 2& 3& 4& 5& 6& 7& 8\\ 5& 7& 6& 3& 4& 8& 1& 2\end{array}\right].$

$($a$)$ Determine $,$ by first rearranging the equation.

$($b$)$ Write each of $,,,$ and $$ in cycle notation.