Read the statement shown below.
“If Jessica is in Grade 8, then Amy is in Grade 9.”
Which of these is logically equivalent to the above statement?
A. If Amy is in Grade 9, then Jessica is not in Grade 8.
B.If Amy is not in Grade 9, then Jessica is not in Grade 8.
C.If Jessica is in Grade 8, then Amy cannot be in Grade 9.
D. If Jessica is not in Grade 8, then Amy is in Grade 9.
The Correct Answer and Explanation is:
The correct answer is:
D. If Jessica is not in Grade 8, then Amy is in Grade 9.
Explanation
To determine the logical equivalence of the given statement, we need to analyze its structure and apply reasoning based on conditional statements and their equivalent forms.
Original Statement:
“If Jessica is in Grade 8, then Amy is in Grade 9.”
This is a conditional statement expressed as ( P \to Q ), where:
- ( P ): Jessica is in Grade 8.
- ( Q ): Amy is in Grade 9.
Logical Equivalence of Conditional Statements:
A conditional statement ( P \to Q ) is logically equivalent to its contrapositive ( \sim Q \to \sim P ), which means:
- If ( Q ) (Amy is in Grade 9) is false, then ( P ) (Jessica is in Grade 8) must also be false.
In simpler terms, if Amy is not in Grade 9, then Jessica is not in Grade 8. This understanding is the key to identifying the correct answer.
Analyzing the Options:
- Option A: “If Amy is in Grade 9, then Jessica is not in Grade 8.”
- This is not logically equivalent. The original statement does not imply this.
- Option B: “If Amy is not in Grade 9, then Jessica is not in Grade 8.”
- This is the inverse of the original statement and is not equivalent to it.
- Option C: “If Jessica is in Grade 8, then Amy cannot be in Grade 9.”
- This directly contradicts the original statement, making it invalid.
- Option D: “If Jessica is not in Grade 8, then Amy is in Grade 9.”
- This is the contrapositive of the original statement and is logically equivalent.
Conclusion:
The contrapositive preserves the truth of the original statement, making Option D the correct answer.