Sure, let's carefully understand the relevant concepts and arrive at the correct answer.
1. Original Conditional Statement:
The given conditional statement is [tex]\( p \rightarrow q \)[/tex]. This means "if [tex]\( p \)[/tex], then [tex]\( q \)[/tex]".
2. Understanding Contrapositive:
The contrapositive of a conditional statement [tex]\( p \rightarrow q \)[/tex] is formed by negating both the hypothesis and the conclusion, and then reversing them. So, the contrapositive of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim q \rightarrow \sim p \)[/tex].
3. Explanation of Choices:
- [tex]\( q \rightarrow p \)[/tex]: This is the converse of the conditional statement, not the contrapositive.
- [tex]\( \sim q \rightarrow \sim p \)[/tex]: This is the negation and reversal of both parts of the original statement, which is the correct formation of the contrapositive.
- [tex]\( p \rightarrow q \)[/tex]: This is the original statement itself, not the contrapositive.
- [tex]\( \sim p \rightarrow \sim q \)[/tex]: This is the inverse of the conditional statement, not the contrapositive.
4. Conclusion:
Based on the conceptual understanding above, the contrapositive of the original conditional statement [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim q \rightarrow \sim p \)[/tex].
Thus, the correct answer is:
[tex]\[
\sim q \rightarrow \sim p
\][/tex]
Therefore, the correct representation of the contrapositive of the conditional statement [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim q \rightarrow \sim p \)[/tex].
