To determine the truth value of the statement [tex]\(q \rightarrow p\)[/tex], we first need to understand the concept of logical implication. The implication [tex]\(q \rightarrow p\)[/tex] is read as "if [tex]\(q\)[/tex], then [tex]\(p\)[/tex]" and is only false in the case where the antecedent [tex]\(q\)[/tex] is true and the consequent [tex]\(p\)[/tex] is false. Otherwise, the implication is true.
Given:
- [tex]\(q\)[/tex] is true
- [tex]\(p\)[/tex] is false
According to the definition of implication [tex]\(q \rightarrow p\)[/tex]:
- If [tex]\(q\)[/tex] is true and [tex]\(p\)[/tex] is false, the implication [tex]\(q \rightarrow p\)[/tex] is false.
Substituting the given truth values:
- [tex]\(q\)[/tex] is true (True)
- [tex]\(p\)[/tex] is false (False)
- The implication [tex]\(q \rightarrow p\)[/tex] is false.
Thus, evaluating [tex]\(q \rightarrow p\)[/tex] (True [tex]\(\rightarrow\)[/tex] False):
- It results in false.
Therefore, the given statement [tex]\(q \rightarrow p\)[/tex] is false because when [tex]\(q\)[/tex] is true and [tex]\(p\)[/tex] is false, the implication [tex]\(q \rightarrow p\)[/tex] is false.
The correct answer is:
B. The statement is false because the truth value of true [tex]\(\rightarrow\)[/tex] false is false.
