To determine the truth value of the compound statement [tex]\(\neg[(\neg p \wedge q) \vee r]\)[/tex], we follow these steps:
1. Identify the truth values for [tex]\(p\)[/tex], [tex]\(q\)[/tex], and [tex]\(r\)[/tex]:
- [tex]\(p\)[/tex] is true ([tex]\(p = \text{True}\)[/tex]).
- [tex]\(q\)[/tex] is false ([tex]\(q = \text{False}\)[/tex]).
- [tex]\(r\)[/tex] is false ([tex]\(r = \text{False}\)[/tex]).
2. Calculate [tex]\(\neg p\)[/tex]:
[tex]\[
\neg p = \neg (\text{True}) = \text{False}
\][/tex]
3. Calculate [tex]\((\neg p \wedge q)\)[/tex]:
[tex]\[
(\neg p \wedge q) = (\text{False} \wedge \text{False}) = \text{False}
\][/tex]
4. Calculate [tex]\((\neg p \wedge q) \vee r\)[/tex]:
[tex]\[
(\neg p \wedge q) \vee r = (\text{False} \vee \text{False}) = \text{False}
\][/tex]
5. Calculate [tex]\(\neg[(\neg p \wedge q) \vee r]\)[/tex]:
[tex]\[
\neg[(\neg p \wedge q) \vee r] = \neg (\text{False}) = \text{True}
\][/tex]
Therefore, the truth value of the compound statement [tex]\(\neg[(\neg p \wedge q) \vee r]\)[/tex] is [tex]\(\text{True}\)[/tex].
