Consider the following statements :
A = If n is even, then n2 is even.
B = If n2 is not even, n is not even.
C = If n2 is even, then n is even.
D = If n is not even, then n2 is not even.
Which of the following statements is true ?
Correct Answer :
C is converse of A.
Solution :
The correct option is C is converse of A.
Let us analyze the given logical statements:
Let statement P be "n is even" and statement Q be "n2 is even".
Then, the original statement A can be written in logical implication form as:
A: If P, then Q (written as P ⇒ Q).
Now, let us define the related logical statements based on statement A (P ⇒ Q):
1. Converse of A:
The converse of an implication (P ⇒ Q) is formed by swapping the hypothesis and the conclusion: "If Q, then P".
Here, Q is "n2 is even" and P is "n is even".
Therefore, the converse of A is: "If n2 is even, then n is even", which is precisely statement C.
2. Inverse of A:
The inverse of an implication (P ⇒ Q) is formed by negating both the hypothesis and the conclusion: "If not P, then not Q".
Here, "not P" is "n is not even" and "not Q" is "n2 is not even".
Therefore, the inverse of A is: "If n is not even, then n2 is not even", which corresponds to statement D.
3. Contrapositive of A:
The contrapositive of an implication (P ⇒ Q) is formed by swapping and negating both the hypothesis and the conclusion: "If not Q, then not P".
Here, "not Q" is "n2 is not even" and "not P" is "n is not even".
Therefore, the contrapositive of A is: "If n2 is not even, then n is not even", which corresponds to statement B.
Comparing these definitions with the given options:
- "C is converse of A." is true.
- "D is converse of A." is false (D is the inverse of A).
- "B is inverse of A." is false (B is the contrapositive of A).
- "D is contraposition of A." is false (D is the inverse of A).
Thus, the true statement among the options is C is converse of A.
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