Minimization using set of equivalent states
First set of equivalent states
$\pi$0 ={ I,II} I={p,q,r} set of non-final state II = {s,t} set of final state
now check states in set I are equivalents or not
p x a -> s [II , goes to state that is in set II] q x a -> t [II] r x a -> r [I]
p x b -> q [I] q x b -> r [I] r x b -> r[I]
it is clear p,q are equivalent (both states on symbol a goes to states that is in set II , both states on symbol b goes to states that is in set I) but r behaves differently, set {p,q,r} divides into two set {p,q},{r}
now check states in set II are equivalent or not
s x a -> s[II] t x a -> t [II]
s x b -> s[II] t x a ->t [II]
so s,t are equivalent states
that results in
Second set of equivalent states
$\pi$1 ={ I,II,III} I = {p,q} II = {r} III = {s,t}
check all states in I are equivalents or not , same for set II and set III using same procedure as above
Third set of equivalent states
$\pi$2 ={ I,II,III,IV} I = {p} II = {q} III = {r} IV = {s,t}
further
$\pi$3 =$\pi$2 .i,e that cannot be further minimized
s and t are equivalent states
Minimized DFA will be same as given in option A