Question
Consider the following two statements : P : If 7 is an odd number, then 7 is divisible by 2. Q : If 7 is a prime number, then 7 is an odd number If V 1 is the truth value of the contrapositive of P and V 2 is the truth value of contrapositive of Q, then the ordered pair (V 1 , V 2 ) equals :
Options
Solution
Key Concepts and Formulas
- Conditional Statement: A statement of the form "If P, then Q," denoted as P → Q.
- Contrapositive: The contrapositive of P → Q is ¬Q → ¬P, where ¬ represents negation. A conditional statement and its contrapositive are logically equivalent (i.e., they have the same truth value).
- Truth Values: A statement can be either true (T) or false (F).
Step-by-Step Solution
Step 1: Find the contrapositive of statement P.
Statement P is: If 7 is an odd number, then 7 is divisible by 2. P can be represented as: (7 is odd) → (7 is divisible by 2). The contrapositive of P is: If 7 is not divisible by 2, then 7 is not an odd number.
Step 2: Determine the truth value of the contrapositive of P.
The contrapositive of P is: If 7 is not divisible by 2, then 7 is not an odd number. 7 is not divisible by 2 is a true statement. 7 is not an odd number is a false statement. Since the hypothesis (7 is not divisible by 2) is true and the conclusion (7 is not an odd number) is false, the contrapositive statement is false. Therefore, V<sub>1</sub> = F.
Step 3: Find the contrapositive of statement Q.
Statement Q is: If 7 is a prime number, then 7 is an odd number. Q can be represented as: (7 is prime) → (7 is odd). The contrapositive of Q is: If 7 is not an odd number, then 7 is not a prime number.
Step 4: Determine the truth value of the contrapositive of Q.
The contrapositive of Q is: If 7 is not an odd number, then 7 is not a prime number. 7 is not an odd number is a false statement. 7 is not a prime number is a false statement. A conditional statement is true if the hypothesis is false, regardless of the truth value of the conclusion. Thus, since the hypothesis (7 is not an odd number) is false, the entire contrapositive statement is true. Therefore, V<sub>2</sub> = T.
Step 5: Determine the ordered pair (V<sub>1</sub>, V<sub>2</sub>).
Since V<sub>1</sub> = F and V<sub>2</sub> = T, the ordered pair (V<sub>1</sub>, V<sub>2</sub>) is (F, T).
Common Mistakes & Tips
- Remember the correct form of the contrapositive: ¬Q → ¬P. It is not the same as the converse or the inverse.
- A conditional statement is only false when the hypothesis is true and the conclusion is false. In all other cases, it's true.
- When negating statements, be careful with quantifiers (e.g., "all," "some," "none").
Summary
We determined the truth values of the contrapositives of statements P and Q. First, we found the contrapositive of each statement by negating and reversing the hypothesis and conclusion. Then, we evaluated the truth value of each contrapositive based on the truth values of its hypothesis and conclusion. The truth value of the contrapositive of P (V<sub>1</sub>) was found to be false, and the truth value of the contrapositive of Q (V<sub>2</sub>) was found to be true. Therefore, the ordered pair (V<sub>1</sub>, V<sub>2</sub>) is (F, T).
Final Answer The final answer is \boxed{(F, T)}, which corresponds to option (C).