To determine which statement is logically equivalent to [tex]\( q \rightarrow p \)[/tex], we need to find the statement that has the same truth value as [tex]\( q \rightarrow p \)[/tex]. This involves identifying the contrapositive of the given conditional statement.
Let's review the given options and analyze each:
1. If a student achieves 90 percent on the geometry final, then the student will pass geometry class.
This statement translates to [tex]\( p \rightarrow q \)[/tex]. It is not logically equivalent to [tex]\( q \rightarrow p \)[/tex], since it is simply the converse of the original statement.
2. If a student passes geometry class, then the student achieved 90 percent on the geometry final.
This statement translates directly to [tex]\( q \rightarrow p \)[/tex]. This is our original conditional statement, but not the statement we are looking for since the problem asks for a logically equivalent statement, which usually refers to a different but logically equivalent form.
3. If a student did not achieve 90 percent on the geometry final, then the student did not pass geometry class.
This statement translates to [tex]\( \neg p \rightarrow \neg q \)[/tex]. This is the contrapositive of [tex]\( q \rightarrow p \)[/tex]. The contrapositive of any conditional statement is always logically equivalent to the original statement.
4. If a student did not pass geometry class, then the student did not achieve 90 percent on the geometry final.
This statement translates to [tex]\( \neg q \rightarrow \neg p \)[/tex]. This is known as the inverse of the original statement. While inverses hold a specific relationship with the original statement, they are not always logically equivalent.
Given this analysis, the statement "If a student did not achieve 90 percent on the geometry final, then the student did not pass geometry class." is the logically equivalent one to [tex]\( q \rightarrow p \)[/tex].
Therefore, the correct answer is:
If a student did not achieve 90 percent on the geometry final, then the student did not pass geometry class.
