$1.$ Let A $=$Washington DC is not the capitol of the U$.$S$.,$ B $=$New York City is the largest city in Massachusetts$,$ and C $=$Ohio is a U$.$S$.$ state$.$ Determine truth value of the following Boolean expression:

a$.$ A $$ B $$C

b$.$ A $->($B $$ C$)$

c$.($A $$ B$)->$ C

d$.($A $($B $$ C$))->(($A $$ B$)($A $$ C$))$

$2.$ Prove that the following logic statements are logically true. Please use proof by contradiction.

a$.(($p$$r$)($q$$r$))(($p$$q$)$r$)$

b$.($p$$q$)($q$$p$)$

c$.($p $+$ q$)($p $+$q$)=0$

$3.$ Please prove the following using truth tables.

$($a $$ b$)($a $$ c$)$ a $$ b $$ c

$($p$$q$)($p$$r$)($p$$r$)($q$$r$)$