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The question is:

Let the predicate $L(x,y)$ stand for "$x$ loves $y$", where the universe is the set $\{cat, dog, fish, bird, snake\}$

Translate the following statements into logical notation:

(a) Everyone loves the cat.

(b) No animal loves any animal.

(c) Some animals are not loved by anyone.

(d) Some animals both love and are loved by the cat.

Translate the following from logical notation to English

(e) $\exists x \forall y L(x,y)$

My attempt:

(a) $\forall x (L(x,cat))$

(b) $\neg \exists x (L(x,y))$

(c) $\exists y(\neg L(x,y)$

(d) $\exists x(L(x,cat) \land L(cat,x))$

(e) Some animals love every animal.

I am trying to ensure I understand this topic, so please correct me if my answers are wrong?

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Hint: two quantifiers in the English, two quantifiers in the translation. Hence, for example, on the more natural reading,

No animal loves any animal

= (No animal x is such that)(There is some animal y such that) $L(x, y)$

= not(There is some animal x such that)(There is some animal y such that) $L(x, y)$

= $\neg\exists x\exists y\;L(x, y)$

Or, equivalently = $\forall x\neg\exists y\;L(x, y)$

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    $\begingroup$ Thank you! Based on your hint I assume (c) is incorrect as well, and the remaining are correct? $\endgroup$ – Shannon Jan 18 '18 at 23:41
  • $\begingroup$ Check also exactly where brackets should go in your syntax. Compare e.g. your (a) with the question (e). $\endgroup$ – Peter Smith Jan 18 '18 at 23:43

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