$($a$)$ Use algebraic manipulations $($Boole$\text{'}$s laws$)$ to show the following logical equivalence:

$(($X $$ Y $)($X $->($Y $$ Z$)))($X $$Y $$ Z$).$

$($b$)$ What is the truth value of the logical formula $((($A$$B$)($A$$B$))->$ C$)$ which corresponds

at the valuation v$($A$)=$ V$,$ v$($B$)=$ V and v$($C$)=$ F$.$

$($c$)$ For the following logical formula, use the method of your choice to determine whether it is

of a tautology, contradiction or contingent formula:

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