The relation between Electric Field and Electric Potential

In your question, V is the absolute potential at the given point while electric field E=potential difference/d.

 

Does any one of them have to be true?

It looks like there are pairs of them which are logically equivalent.

 

Te electric field E and electric potential V are two separate functions. Although the potential depends on the electric field, they are not proportional and the potential depends on the integral of the electric field over a path. The forum rules don't allow simply giving the answer, but the answer is quite simple. @SammyS The pairs are not logically equivalent. The equation the OP presents that E=V/d does have precise proportionality between E and V, but this equation is very misleading because it does not apply in general. It is for the special case of an ideal capacitor and E is the uniform electric field between the plates and V is the voltage drop across the plates. The equation really does not apply here, and the capacitor equation does not give V at any location between the plates where the E field is present. Although it looks like the right equation, it is totally irrelevant to this problem.

 

Are you saying that there is no pair that are not logically equivalent?

From a pure logic point of view.

(P implies Q) is logically equivalent to ((not Q) implies (not P)) .

It appears to me that we can find cases where one of these statements is the contrapositive of another.

 

@SammyS It is somewhat difficult to answer your question without giving out the complete answer (at least what I am pretty certain is the correct answer), but none of the statements contains logical equivalence. To just give a counterexample for statement "d", a charged hollow conducting sphere has E=0 throughout the entire interior, but V is not equal to zero.... editing.. And to give the OP something that might help them answer "b" and "c", what is the V and E for the point midway between two electrical charges of +Q and -Q? And I think the example I gave for "d" can also be used to answer "a".