To determine the logically equivalent statement to [tex]\( p \rightarrow \sim q \dot{q} \)[/tex], follow these steps:
1. Understand the original statement:
[tex]\( p \rightarrow \sim q \dot{q} \)[/tex]
Here, [tex]\( \sim q \dot{q} \)[/tex] means [tex]\( \sim(q \land q) = \sim q \)[/tex].
2. Evaluate the internal logic:
The statement [tex]\( q \land q \)[/tex] is always equal to [tex]\( q \)[/tex]. Therefore, [tex]\( \sim(q \land q) \)[/tex] simplifies to [tex]\( \sim q \)[/tex].
3. Rewrite original statement:
The given statement thus simplifies to [tex]\( p \rightarrow \sim q \)[/tex].
4. Determine the equivalent form:
To find the equivalent form, let’s compare the simplified statement [tex]\( p \rightarrow \sim q \)[/tex] with the given logical options.
- [tex]\( p \rightarrow q \)[/tex]: This suggests that if [tex]\( p \)[/tex] is true, then [tex]\( q \)[/tex] must be true. This is different from our simplified statement [tex]\( p \rightarrow \sim q \)[/tex].
- [tex]\( \sim p \rightarrow q \)[/tex]: This suggests that if [tex]\( p \)[/tex] is not true, then [tex]\( q \)[/tex] is true. This does not match [tex]\( p \rightarrow \sim q \)[/tex] either.
- [tex]\( q \rightarrow p \)[/tex]: This means if [tex]\( q \)[/tex] is true, then [tex]\( p \)[/tex] must be true. This is also not equivalent to [tex]\( p \rightarrow \sim q \)[/tex].
- [tex]\( q \rightarrow \sim p \)[/tex]: This means if [tex]\( q \)[/tex] is true, then [tex]\( p \)[/tex] must be false. This form can be rewritten in its contrapositive form as [tex]\( p \rightarrow \sim q \)[/tex], which matches exactly.
Thus, the logically equivalent statement to [tex]\( p \rightarrow \sim q \dot{q} \)[/tex] is:
[tex]\[ q \rightarrow \sim p \][/tex]
Therefore, the correct choice is:
[tex]\[ q \rightarrow \sim p \][/tex]
