To determine the logically equivalent statement to the conditional statement [tex]\( p \rightarrow \sim q \)[/tex], we can follow a series of logical equivalences and reasoning.
1. Understanding the Implication:
The statement [tex]\( p \rightarrow \sim q \)[/tex] means "if [tex]\( p \)[/tex] then not [tex]\( q \)[/tex]".
2. Using Logical Equivalences:
There's a fundamental equivalence in logic that can help us:
[tex]\[
p \rightarrow q \text{ is equivalent to } \sim q \rightarrow \sim p
\][/tex]
This suggests that the implication can be rephrased using its contrapositive. However, instead of directly using this equivalence, we will derive the equivalent form.
3. Rewriting the Condition:
The given statement, [tex]\( p \rightarrow \sim q \)[/tex], can be interpreted as:
- If [tex]\( p \)[/tex] is true, then [tex]\( q \)[/tex] must be false.
4. Finding the Contrapositive:
The contrapositive of the statement [tex]\( p \rightarrow \sim q \)[/tex] is found by negating both parts and swapping them. This means we take:
- [tex]\(\sim(\sim q)\)[/tex], which simplifies to [tex]\( q \)[/tex]
- [tex]\(\sim p\)[/tex]
So, the contrapositive becomes:
[tex]\[
\sim q \rightarrow \sim p \text{ is equivalent to } q \rightarrow \sim p
\][/tex]
5. Conclusion:
Based on our logical equivalence transformation, the statement [tex]\( p \rightarrow \sim q \)[/tex] is logically equivalent to [tex]\( q \rightarrow \sim p \)[/tex].
Therefore, the logically equivalent statement to [tex]\( p \rightarrow \sim q \)[/tex] is:
\[
q \rightarrow \sim p
\
