### Artificial IntelligenceAIMA Exercises

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 $A$, $B$, $C$.

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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