Question: 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:

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> No one who can ski plays soccer
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> 
> 

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

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> 1. ∀x(¬(P(x)∧H(x))
> 2. ∀*x* (H(*x*) -> ~p(*x*))
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> 
> 

[Symbols Reference](https://en.wikipedia.org/wiki/List_of_logic_symbols)

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

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> ¬∃xH(x) ∧ P(x)
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> 
> ≡ ∀x¬(H(x) ∧ P(x)) De Morgan's Law
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> 
> ≡ ∀x(¬H(x) ∨ ¬P(x)) De Morgan's Law
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> 
> ≡ ∀x(H(x) → ¬P(x)) Implication Relation
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> 
> ≡ ∀x(P(x) → ¬H(x)) Contrapositive
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> 
> 

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