I've just started learning propositional and predicate logic. One of the tasks I have is to find a domain B, and predicates D(x) and E(x) such that these two conditions are met:
The statement "There exists an A, element of set B such that not D(A) -> There exists an A, element of set B, such that not E(A)" is true.
The statement "There exists an A, element of set B, such that not D(A) -> not E(A)." is false.
What I don't understand is how these two conditions can be met at the same time. To my understanding, P -> Q will return True no matter what Q's value is, if P is false. Hence, for "There exists an A, element of set B, such that not D(A) -> not E(A)." to be false, D(A) must return True for all A.
Given this, for "There exists an A, element of set B such that not D(A) -> There exists an A, element of set B, such that not E(A)" to be true, not E(A) must be true. So I've shown there exists an A such that not D(A) is true, and not E(A) is true. In that case, then wouldn't that mean "There exists an A, element of set B, such that not D(A) -> not E(A)." is true? So how can 1) be true, and 2) be false for any domain B, and predicates D(x) and E(x)?
Please don't give me an answer/hint. I actually want to solve this myself. I'm just hoping someone could explain to me where my understanding of predicate and propositional logic has gone wrong? Because to my current understanding, this question is unsolvable. These two conditions can't be met at the same time.