1. (∀x)(∃y)Axy ⊃ (∀x)(∃y)Bxy
2. (∃x)(∀y)∼Bxy
-Which of the following propositions is derivable from the given premises in F?
A) (∃x)(∀y)∼Axy
B) (∃x)(∀y)~Ayx
C) (∀x)(∃y)Axy
D) (∀x)(∃y)Ayx
E) (∃x)(∃y)(Axy • Bxy)

Question

Quizplus · Accepted Answer

BFrom the first premise, we understand that for every x, there exists a y such that Axy implies that for every x, there exists a y such that Bxy. The second premise states that there exists an x for which all y do not satisfy Bxy. This situation makes it impossible for every x to have a corresponding y that satisfies Bxy, as required by the implication of the first premise, unless there is an x for which all y do not satisfy Axy as well, making it impossible to trigger the condition of the implication. Therefore, both A and B are derivable because they express the existence of an x for which all y do not satisfy Axy or Ayx, respectively, which is a necessary condition to prevent a contradiction with the second premise.

1. (∀X)(∃y)Axy ⊃ (∀X)(∃y)Bxy