Let $\displaystyle C=\{a,b,c,d,e,f,g,h\}$. Let $\displaystyle R=\{(f,g),(h,a),(d,e),(e,e),(g,h),(b,c),(b,f),(c,g )\}$ be a relation on the set $\displaystyle C$. Use the matrix representation of relations in order to compute the following:

a) Let $\displaystyle S$ be the reflexive closure of $\displaystyle R$. Compute $\displaystyle S$.

My current answer is $\displaystyle S=\left(\begin{array}{cccccccc}1&0&0&0&0&0&0&0\\0& 1&1&0&0&1&0&0\\0&0&1&0&0&0&1&0\\0&0&0&1&1&0&0&0\\0 &0&0&0&1&0&0&0\\0&0&0&0&0&1&1&0\\0&0&0&0&0&0&1&1\\ 1&0&0&0&0&0&0&1\end{array}\right)$

b) Let $\displaystyle T$ be the transitive closure of $\displaystyle R$. Compute $\displaystyle T$. (I used Warshall's algorithm)

My current answer is $\displaystyle T=\left(\begin{array}{cccccccc}0&0&0&0&0&0&0&0\\0& 0&1&0&0&1&0&0\\1&0&1&0&0&0&1&1\\0&0&0&0&1&0&0&0\\0 &0&0&0&1&0&0&0\\1&0&0&0&0&1&1&1\\1&0&0&0&0&0&0&1\\ 1&0&0&0&0&0&0&0\end{array}\right)$

c) Let $\displaystyle D=S\cup T$. Compute the symmetric closure of $\displaystyle D$ and call this set $\displaystyle F$.

My current answer is $\displaystyle F=\left(\begin{array}{cccccccc}1&0&1&0&0&1&1&1\\0& 1&1&0&0&1&0&0\\1&1&1&0&0&0&1&1\\0&0&0&1&1&0&0&0\\0 &0&0&1&1&0&0&0\\1&1&0&0&0&1&1&1\\1&0&1&0&0&1&1&1\\ 1&0&1&0&0&1&1&1\end{array}\right)$

d) Prove or disprove that $\displaystyle F$ is an equivalence relation on the set $\displaystyle C$. If $\displaystyle F$ is an equivalence relation, then list the elements in each of the equivalence classes. If $\displaystyle F$ is not an equivalence relation, then list what elements need to be added to $\displaystyle F$ to make it an equivalence relation.

(I have not completed this part yet, so I could use a hand here)

I mainly need to know whether my work on the first three parts has yielded correct answers (it'd be pretty easy to make a mistake). As for the last part, any help would be appreciated.