$6.[15\%]$ From equation to state graph.

A sequential circuit has three D flip$-$flops, A$,$ B$,$ C$,$ one input X$,$ andone output Z$.$ The flip$-$flop input equations are described as follows:

A$*=$ DA $=$ A'BC' $+$ XA'B'C $+$X'AB'C'

B$*=$ Ds $=$ X'A'C $+$ XA'C'

C$*=$ Dc$=$ X'A'B' $+$ X'A'C $+$ A'B'C

$($a$)[3\%]$ Please derive the state table for the design. $($assume So $=$

$"000",$ S$1="001",$ S$2="010",$ S$3="011",$ S$4="100",$ Ss $="101",$ Ss $=$

$"110",$ S$7="111")$

$($b$)[4\%]$ Please draw the state graph if the output equation is: Z $=$X'A'BC $+$ X'AB'C' $+$ XA'BC $+$ XAB'C'

$($c$)[4\%]$ Please draw the state graph if the output equation is: Z $=$AB'C' $+$ A'BC

$($d$)[4\%]$ Continue on $($c$),$ can you use only $2$ state variables to

implement a state machine which have the same function? If so$,$

please show how you eliminate the redundant states, and drawthe state graph with minimum number of states.