select the best translation into predicate logic.
-Some athletes who don't work hard receive scholarships, if, and only if, no athletes who play professional sports don't have determination.
A) (∃x)[(Ax • ∼Wx) • Sx] ≡ (∀x)[(Ax • Px) ⊃ ∼∼Dx]
B) (∃x)[(Ax • ∼Wx) • Sx] ⊃ (∀x)[(Ax • Px) ⊃ ∼∼Dx]
C) (∃x)[(Ax • ∼Wx) • Sx] ⊃ (∀x)[(Ax • Px) ⊃ ∼Dx]
D) (∃x)[(Ax • ∼Wx) • Sx] ≡ (∀x)[(Ax • Px) ⊃ ∼Dx]

Question

Quizplus · Accepted Answer

The correct translation is A) because it accurately represents the given statement using predicate logic. The left side of the equivalence (≡) states that there exists at least one athlete who doesn't work hard and receives a scholarship. The right side of the equivalence states that for all athletes, if they play professional sports, then they have determination (noted by the double negation ∼∼Dx which simplifies to Dx). This matches the structure of "if, and only if" from the original statement.

Select the Best Translation into Predicate Logic