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?
Best Answer
Yes, you have the basic form of: $\exists x \,\exists y\,\Big(\big(x\neq y\big)\wedge Q(x)\wedge Q(y)\Big)$
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.