My friend and I were looking over my homework and he pointed out something that he thought was incorrect.
I was to write sentances using logical connectives.
The original sentance was: "To get an A in this class, it is necessary for you to get an A on the final exam.
I figured the translation would be:
Let p = You get an A in this class.
Let q = You get an A on the final.
q -> p
But he figured the correct way to write the sentance would be the inverse:
p -> q
I agreed, but as I continued in my book I found an example the seemed to confirm what I had originally thought. Could anyone help me sort out who is correct?
Thank you in advance.
Your friend is correct.
To see why, consider the implication $q\implies p$. This states that if you get an A on the final exam, then you get an A in the class. This is clearly not what is meant by the original sentence. The original sentence says instead that you must get an A on the final to get an A in the class. Therefore anyone who receives an A in the class must have received an A on the final exam. This is precisely the meaning of $p \implies q$.
Another way to see how your friend is correct is to remember that the contrapositive of a statement shares its truth value. Thus, if the implication $q \implies p$ is true then so too is its contrapositive $!p \implies !q$. This implication states if you did not get an A in the class, then you did not get an A on the final, which is also clearly not the meaning of the original sentence because while getting an A on the final is necessary it is not a sufficient condition for receiving an A in the course.