$1.$ Determine whether each of these compound propositions is satisfiable.

$($a$)($p$$q$$r$)($p$$q$$s$)($p$$r$$s$)($p$$q$$s$)($p$$q$$s$)$

$($b$)($p $$q $$ r$)($p $$ q $$s$)($p $$q $$s$)($p $$r $$s$)($p $$ q $$r$)($p $$r $$s$)$

$($c$)($p$$q $$r$)($p$$q $$s$)($q $$r $$s$)($p$$ r $$s$)($p$$q $$s$)($p $$q $$r$)($p $$q $$ s$)($p $$r $$s$)$

$2.$ If p$1,$ p$2,...,$ pn are n propositions, explain why n$^1$ i$=1^$n j$=$i$+1($pi $$pj $)$ is true if and only if at most one of p$1,$ p$2,...,$ pn is true.

$3.$ An ancient Sicilian legend says that the barber in a remote town who can be reached only by traveling a dangerous mountain road shaves those people, and only those people, who do not shave themselves. Can there be such a barber?

$4.$ Find the solution to each of these recurrence relations with the given initial conditions. Use an iterative approach. $($a$)$ an $=$ an$1$ n$,$ a$0=4($b$)$ an $=2$an$13,$ a$0=1$