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Using predicate logic where the pets are cats, dogs, and parrots ($C(x)$, $D(x)$, and $P(x)$ respectively). How would you translate the sentence: There are at least two students who have all three pets. Where the domain is all students in class.

I came to the answer: $ \exists x \exists y ~ ((P(x) \land D(x) \land C(x) \land (D(y) \land P(y) \land C(y) \land (x \ne y))) $.

This can be read as for some $x$, $x$ has all three pets and for some $y$ that is not $x$ has all three pets.

Can someone please tell me if this is correct? I think it is wrong. Don't I have to say something about the rest of the domain besides just these two?

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    $\begingroup$ It looks correct to me $\endgroup$ – Guest Sep 9 '15 at 23:45
  • $\begingroup$ You only care about $x$ and $y$, it's okay that you don't say anything about the rest of the domain $\endgroup$ – shost71 Sep 10 '15 at 0:06
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Yes, you have the basic form of: $\exists x \,\exists y\,\Big(\big(x\neq y\big)\wedge Q(x)\wedge Q(y)\Big)$

  • There are (at least) two things which are distinct but both have the required quality.

don't i have to say something about the rest of the domain besides just these two??

No.   Why do you thing so?   There might be students who don't own all three pets, there might be only students who own all they pets, or there might only be two students in the class.   We don't know, and don't care.   Only the existence and distinctness of two students who own all three pets needs be asserted.

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