$7.[20$ pts$]$ Consider the system in Figure $4$ and suppose the health of gates $\backslash ($ X$,$ Y $\backslash )$ and $\backslash ($ Z $\backslash )$ are represented by variables $\backslash ($ o k X$,$ o k Y $\backslash )$ and $\backslash ($ o k Z $\backslash ).$

$(5$pt$)$ Write the system description as a CNF$.$

Suppose the system input is $\backslash ($ A$=1,$ B$=0\backslash )$ and the system output is $\backslash ($ E$=0\backslash ).$ Answer the following questions under this system observation.

$(8$pt$)$ Construct the health condition for the system using directed resolution.

$(4$pt$)$ List the kernel diagnoses.

$(3$pt$)$ List the minimal$-$cardinality diagnoses.