Exercise 7.5

Which of the following are correct?

  1. ${False} \models {True}$.

  2. ${True} \models {False}$.

  3. $(A\land B) \models (A{\;\;{\Leftrightarrow}\;\;}B)$.

  4. $A{\;\;{\Leftrightarrow}\;\;}B \models A \lor B$.

  5. $A{\;\;{\Leftrightarrow}\;\;}B \models \lnot A \lor B$.

  6. $(A\lor B) \land (\lnot C\lor\lnot D\lor E) \models (A\lor B\lor C) \land (B\land C\land D{:\;{\Rightarrow}:\;}E)$.

  7. $(A\lor B) \land (\lnot C\lor\lnot D\lor E) \models (A\lor B) \land (\lnot D\lor E)$.

  8. $(A\lor B) \land \lnot(A {:\;{\Rightarrow}:\;}B)$ is satisfiable.

  9. $(A\land B){:\;{\Rightarrow}:\;}C \models (A{:\;{\Rightarrow}:\;}C)\lor(B{:\;{\Rightarrow}:\;}C)$.

  10. $(C\lor (\lnot A \land \lnot B)) \equiv ((A{:\;{\Rightarrow}:\;}C) \land (B {:\;{\Rightarrow}:\;}C))$.

  11. $(A{\;\;{\Leftrightarrow}\;\;}B) \land (\lnot A \lor B)$ is satisfiable.

  12. $(A{\;\;{\Leftrightarrow}\;\;}B) {\;\;{\Leftrightarrow}\;\;}C$ has the same number of models as $(A{\;\;{\Leftrightarrow}\;\;}B)$ for any fixed set of proposition symbols that includes $A$, $B$, $C$.

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