Decide whether each of the following sentences is valid, unsatisfiable, or neither. Verify your decisions using truth tables or the equivalence rules of Table logical-equivalence-table (page logical-equivalence-table). 1. ${Smoke} {\:\;{\Rightarrow}\:\;}{Smoke}$
2. ${Smoke} {\:\;{\Rightarrow}\:\;}{Fire}$
3. $({Smoke} {\:\;{\Rightarrow}\:\;}{Fire}) {\:\;{\Rightarrow}\:\;}(\lnot {Smoke} {\:\;{\Rightarrow}\:\;}\lnot {Fire})$
4. ${Smoke} \lor {Fire} \lor \lnot {Fire}$
5. $(({Smoke} \land {Heat}) {\:\;{\Rightarrow}\:\;}{Fire}) {\;\;{\Leftrightarrow}\;\;}(({Smoke} {\:\;{\Rightarrow}\:\;}{Fire}) \lor ({Heat} {\:\;{\Rightarrow}\:\;}{Fire}))$
6. $({Smoke} {\:\;{\Rightarrow}\:\;}{Fire}) {\:\;{\Rightarrow}\:\;}(({Smoke} \land {Heat}) {\:\;{\Rightarrow}\:\;}{Fire}) $
7. ${Big} \lor {Dumb} \lor ({Big} {\:\;{\Rightarrow}\:\;}{Dumb})$