Prove each of the following assertions:
1. $\alpha$ is valid if and only if ${True}{\models}\alpha$.
2. For any $\alpha$, ${False}{\models}\alpha$.
3. $\alpha{\models}\beta$ if and only if the sentence $(\alpha {\:\;{\Rightarrow}\:\;}\beta)$ is valid.
4. $\alpha \equiv \beta$ if and only if the sentence $(\alpha{\;\;{\Leftrightarrow}\;\;}\beta)$ is valid.
5. $\alpha{\models}\beta$ if and only if the sentence $(\alpha \land \lnot \beta)$ is unsatisfiable.
1. $\alpha$ is valid if and only if ${True}{\models}\alpha$.
2. For any $\alpha$, ${False}{\models}\alpha$.
3. $\alpha{\models}\beta$ if and only if the sentence $(\alpha {\:\;{\Rightarrow}\:\;}\beta)$ is valid.
4. $\alpha \equiv \beta$ if and only if the sentence $(\alpha{\;\;{\Leftrightarrow}\;\;}\beta)$ is valid.
5. $\alpha{\models}\beta$ if and only if the sentence $(\alpha \land \lnot \beta)$ is unsatisfiable.
Prove each of the following assertions:
1. $\alpha$ is valid if and only if ${True}{\models}\alpha$.
2. For any $\alpha$, ${False}{\models}\alpha$.
3. $\alpha{\models}\beta$ if and only if the sentence
$(\alpha {\:\;{\Rightarrow}\:\;}\beta)$ is valid.
4. $\alpha \equiv \beta$ if and only if the sentence
$(\alpha{\;\;{\Leftrightarrow}\;\;}\beta)$ is valid.
5. $\alpha{\models}\beta$ if and only if the sentence
$(\alpha \land \lnot \beta)$ is unsatisfiable.