Question
Answers

If P implies Q, an equivalent statement is
(A) Q implies P
(B) Q is a necessary condition for P
(C) P is a necessary condition for Q
(D) Not P implies Q
(E) Not P implies not Q

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Hint: We have the statement “ If P implies Q”. The statement P implies Q means that the logical meaning of P is dependent on the logical meaning of Q. In other words, we can say that P can only be true if and only if Q is true or P can be false if and only if Q is false. We know that the equivalent statements are those statements that are written differently but have the same logical meaning. Now, conclude the equivalent statement.

Complete step-by-step solution -
According to the question, we have a statement,
If P implies Q ………………………..(1)
We have to find the equivalent statement of the statement in equation (1).
We know that the equivalent statements are those statements that are written differently but have the same logical meaning.
It means we have to find a statement which has the same meaning as the statement “ If P implies Q ”.
Now, from equation (1), we have the statement that if P implies Q. With the help of Q we can say about P.
It means that the logical meaning of P is dependent on the logical meaning of Q. In other words, we can say that,
 “P can only be true if and only if Q is true” ………………………….(2)
Also, “ P can be false if and only if Q is false” ………………………….(3)
From equation (2) and equation (3), we can say that
“P cannot be true unless Q” is true or “ if Q is false, then P is false”.
So, we have the statement that Q is a necessary condition for P.
Therefore, the equivalent statement of the given statement is “ Q is a necessary condition for P”.
Hence, option (B) is the correct one.

Note: In this question, one can think that the meaning of P implies Q is the same as the logical meaning of Q is dependent on the logical meaning of P. This is wrong because the statement P implies Q means that the logical meaning of P is dependent on the logical meaning of Q. Therefore, to rectify this mistake, we have to keep this point in our mind.

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