Let's analyze the given conditional statement [tex]\( p \rightarrow q \)[/tex] and determine which of the provided statements is logically equivalent.
1. Original Statement [tex]\( p \rightarrow q \)[/tex]:
- This means "If [tex]\( p \)[/tex] is true, then [tex]\( q \)[/tex] must be true."
2. Contrapositive:
- The contrapositive of a conditional statement [tex]\( p \rightarrow q \)[/tex] is [tex]\(\sim q \rightarrow \sim p \)[/tex]. This means "If [tex]\( q \)[/tex] is not true, then [tex]\( p \)[/tex] is not true". The contrapositive of a statement is always logically equivalent to the original statement.
3. Options Analysis:
- Option 1: [tex]\( \sim p \rightarrow \sim q \)[/tex]:
- This means "If [tex]\( p \)[/tex] is not true, then [tex]\( q \)[/tex] is not true." This is not the contrapositive of [tex]\( p \rightarrow q \)[/tex], thus, not logically equivalent.
- Option 2: [tex]\( \sim q \rightarrow \sim p \)[/tex]:
- This means "If [tex]\( q \)[/tex] is not true, then [tex]\( p \)[/tex] is not true." This is the contrapositive of the original statement [tex]\( p \rightarrow q \)[/tex], so it is logically equivalent.
- Option 3: [tex]\( q \rightarrow p \)[/tex]:
- This means "If [tex]\( q \)[/tex] is true, then [tex]\( p \)[/tex] must be true." This is the converse of the original statement, not logically equivalent.
- Option 4: [tex]\( p \rightarrow \sim q \)[/tex]:
- This means "If [tex]\( p \)[/tex] is true, then [tex]\( q \)[/tex] is not true." This is not the contrapositive of [tex]\( p \rightarrow q \)[/tex], thus, not logically equivalent.
Therefore, the statement [tex]\( \sim q \rightarrow \sim p \)[/tex] is logically equivalent to [tex]\( p \rightarrow q \)[/tex].
Thus, the correct answer is the second option: [tex]\( \sim q \rightarrow \sim p \)[/tex].
