Translating an English sentence into predicate logic

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I have a question where I was given the following atomic propositions:

Let H(x) = x can ski

Let P(x) = x plays soccer

Note: The universe of discourse is all humans

I was tasked to translate the following sentence logically:

No one who can ski plays soccer

I came up with two solutions for this sentence and I'm unsure if one is considered more correct:

1. ∀x(¬(P(x)∧H(x))

2. ∀x (H(x) -> ~p(x))

Symbols Reference

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2 Answers

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Your second sentence can be rescued by removing the negation at the beginning: $$\forall x (H(x)\to\neg P(x)).$$ This translates to, For every person, if they ski, they don't play soccer. This is equivalent in English to saying that nobody who skis plays soccer.

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According to my professor the following are equivalent ways to express no one who can ski plays soccer in logic.

¬∃xH(x) ∧ P(x)

≡ ∀x¬(H(x) ∧ P(x)) De Morgan's Law

≡ ∀x(¬H(x) ∨ ¬P(x)) De Morgan's Law

≡ ∀x(H(x) → ¬P(x)) Implication Relation

≡ ∀x(P(x) → ¬H(x)) Contrapositive

Thus, both my proposed options in the current version of this question are correct!

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