Let's start by analyzing the given conditional statement: [tex]\(\sim p \rightarrow q\)[/tex].
In propositional logic, an important concept is that of logical equivalence. Two statements are logically equivalent if they have the same truth value in all possible scenarios. One key principle in propositional logic is that a conditional statement is logically equivalent to its contrapositive.
The contrapositive of a conditional statement [tex]\(p \rightarrow q\)[/tex] is [tex]\(\sim q \rightarrow \sim p\)[/tex]. These two statements are always logically equivalent.
Given the statement [tex]\(\sim p \rightarrow q\)[/tex], we need to find the statement that is logically equivalent to it.
Applying the principle of contrapositive, we transform [tex]\(\sim p \rightarrow q\)[/tex] to its contrapositive form:
1. Reverse the hypothesis and conclusion. The hypothesis (antecedent) here is [tex]\(\sim p\)[/tex] and the conclusion (consequent) is [tex]\(q\)[/tex].
2. Negate both the hypothesis and conclusion in the contrapositive. Thus, we negate [tex]\(q\)[/tex] to get [tex]\(\sim q\)[/tex] and [tex]\(\sim p\)[/tex] to get [tex]\(p\)[/tex].
This gives us:
[tex]\[
\sim q \rightarrow p
\][/tex]
Therefore, the statement that is logically equivalent to [tex]\(\sim p \rightarrow q\)[/tex] is:
[tex]\[
\sim q \rightarrow p
\][/tex]
Thus, the correct answer is:
[tex]\[
\sim q \rightarrow p
\][/tex]
