Question

Which of the following is NOT equivalent to $p\to q$
(A) p only if q
(B) q is necessary for p
(C) q only if p
(D) p is sufficient for q

Answer 1

We solve this problem by first going through the definition of the implication. Then we consider all the types in which the conditional implication can be stated. Then we consider those types and interpret their meanings and find which of the given options are correct definitions of given implications $p\to q$. Then the remaining option left out is our required answer.

Complete step-by-step solution:
The given expression is $p\to q$. It is a logical Implication statement.
Now let us discuss the definition of the Implication.
The expression $p\to q$ can be called as “p implies q”. It means if p is true then q is true.
It is also called a conditional statement.
We can also call $p\to q$ as “q if p” and “p only if q”.
As we said above that $p\to q$ can be described as “if p is true then q is true” that means if q is not true then p is not true. So, we can say that q is necessary for p.
Similarly, as we have discussed above, $p\to q$ can be described as “if p is true then q is true”. But we don’t know whether q is true or false when p is false. So, we can say that p is sufficient for q.
So, we get that options (A) (B), and (D) are correct definitions of $p\to q$.
So, only option (C) is not equivalent to $p\to q$.
Hence the answer is Option C.

Note: The common mistake one makes while solving this problem is one might get confused with the definition of the implication $p\to q$, that is one might make a mistake by using the definition of implication as “q only if p” instead of “p only if q”. But it is wrong because “q only if p” is the equivalent form of $q\to p$ not for $p\to q$.