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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. ${Big} \lor {Dumb} \lor ({Big} {:;{\Rightarrow}:;}{Dumb})$