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Let [tex]\( p \)[/tex] represent a true statement, while [tex]\( q \)[/tex] and [tex]\( r \)[/tex] represent false statements.

Find the truth value of the compound statement:
[tex]\[
\sim[(\sim p \wedge q) \vee r]
\][/tex]

Sagot :

GinnyAnswer
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].
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