To determine the truth value of the statement [tex]\((\sim p \vee \sim q) \wedge (\sim r \wedge q)\)[/tex], we need to evaluate each component of the logical expression given the truth values of [tex]\(p\)[/tex], [tex]\(q\)[/tex], and [tex]\(r\)[/tex].
a) When [tex]\(p\)[/tex] is true, [tex]\(q\)[/tex] is true, and [tex]\(r\)[/tex] is true:
- [tex]\(\sim p\)[/tex], which means "not [tex]\(p\)[/tex]", is false because [tex]\(p\)[/tex] is true.
- [tex]\(\sim q\)[/tex], which means "not [tex]\(q\)[/tex]", is false because [tex]\(q\)[/tex] is true.
- [tex]\(\sim r\)[/tex], which means "not [tex]\(r\)[/tex]", is false because [tex]\(r\)[/tex] is true.
- Now, let's evaluate [tex]\(\sim p \vee \sim q\)[/tex]:
- [tex]\(\sim p \vee \sim q\)[/tex] is "false or false", which is false.
- Next, let's evaluate [tex]\(\sim r \wedge q\)[/tex]:
- [tex]\(\sim r \wedge q\)[/tex] is "false and true", which is false.
- Finally, let's evaluate the entire statement [tex]\((\sim p \vee \sim q) \wedge (\sim r \wedge q)\)[/tex]:
- [tex]\((\sim p \vee \sim q) \wedge (\sim r \wedge q)\)[/tex] is "false and false", which is false.
So, the truth value of the statement [tex]\((\sim p \vee \sim q) \wedge (\sim r \wedge q)\)[/tex] when [tex]\(p\)[/tex] is true, [tex]\(q\)[/tex] is true, and [tex]\(r\)[/tex] is true is:
False
