Exercise 2.2.
Let’s consider the interpretation $v$ where $v(p)=F, v(q)=T, v(r)=T$.
Does $v$ satisfy the following propositional formulas?

$(p \rightarrow \neg q) \vee \neg(r \wedge q)$

$(\neg p \vee \neg q) \rightarrow(p \vee \neg r)$

$\neg(\neg p \rightarrow \neg q) \wedge r$

$\neg(\neg p \rightarrow q \wedge \neg r)$

Solution.
v satisfies 1., 3. and 4 .
$v$ doesn’t satisfy 2 .

Exercise 2.1.
Which of the following are well formed propositional formulas?

$\vee p q$

$(\neg(p \rightarrow(q \wedge p)))$

$(\neg(p \rightarrow(q=p)))$

$(\neg(\diamond(q \vee p)))$

$(p \wedge \neg q) \vee(q \rightarrow r)$

$p \neg r$

Solution.
Well formed formulas: 2 . and 5 .

Exercise 2.12. 布
Let’s consider a propositional language where

p means ” $x$ is a prime number”,

$q$ means ” $x$ is odd”.
Formalize the following sentences:

” $x$ being prime is a sufficient condition for $x$ being odd”

” $x$ being odd is a necessary condition for $x$ being prime”

Solution. 1. and 2. $\quad p \rightarrow q$

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