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I wanted to know if what I did is even on the correct path for how this question is worded. How can you have two variables when it's dealing with a single unhappy person? I'm guessing the third way will just be using De Morgan’s Laws for quantifiers?

Translate each of these statements into logical expressions in three different ways by varying the domain and by using predicates with one and with two variables.

  • There is a person in your school who is not happy.

My first solution:

S(x) statement “x is in my school” and H(x) statement “x is happy”
∃x(S(x) ∧ ¬H(x))

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up vote 1 down vote accepted

You might want to say that $\exists x\in \hbox{Human Race}$ s.t. $S(x)\wedge \sim H(x)$. What you have is basically correct. Just bound the $x$ in some set to give context.

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What did it mean by the using predicates with one and with two variables? It sounded like it wanted something like ∃x∀y(statements) – LF4 Jul 6 '12 at 1:49
The clause "$x \in \text{Human Race}$" is a set-theoretical, not strictly speaking logical. (Typically $\in$ is not admitted as a logical symbol) – David Harris Jul 6 '12 at 3:07
All logical arguments must somehow be bounded by a universe of discourse. You could say $\exists \hbox{a human} x$ s.t.... – ncmathsadist Jul 6 '12 at 21:11

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