Please complete these exercises by filling in the blanks with the correct truth-values. You should find the first four sequents to be valid, and the next four invalid. There might be more than one correct answer even though the answer provided here allows only one of them. Sorry about this.

( P  ( Q &  ( R S )))  P ~ S ~  ( Q & R )
(  ( R  ( P ~ Q )))  ~  ( R  ( P Q ))   Q
( ~ ( P & Q &  ~ ( ~ P ~ Q )) ~ ( ~ ( P Q  ( ~ P & ~ Q ))
~ (( ~ P & Q  ( R S ))  ( ~ ( ~ P ~ Q )  &  ~  ( R S ))  ( ~  (( R S ) & ~ P  ( P  ( R S )))
 
((( P Q R  (( ~ P & ~ Q ) R ))  ( P Q )
( P  ( Q & ~ R )) ( ~ P ~ ( ~ Q ~ S ))  ( S ( ~ P T ))
   
( ~ R  →  ~ Q (( ~ P & R & ~ Q ~ ( P  ↔  ( ~ R Q ))
(( P Q )  →  ( ~ P Q ))  ( ~ ( P Q )  &  ( ~ P & ~ Q ))  ~ ( ~ (( P Q )  &  ( P ~ Q ))   ( P Q ))