Re-writing "Everybody loves somebody" and variants in symbolic logic. Right now I'm working on a set of questions and I came across a two-parter that confused me a bit with the way it was asked. The question is simply put as such: Write the statements below in symbolic logic. a. Everybody loves somebody. b. Somebody loves everybody. Would the way to rewrite this be a simple as using existential and universal quantifiers? Such that a. would be translated to "∀x, where x is a person, ∃ person x they love." And b. would be translated to "∃x such that x loves ∀x." Is this a proper way to answer the question? Any help is appreciated.

You have the right ideas, basically, but there is some more work to get the statements into symbolic logic.

Let $L(x,y)$ be the relation "$x$ loves $y$". Then we get:

$$a.\quad ∀x \, ∃y \, L(x,y)$$

$$b.\quad ∃x \, ∀y \, L(x,y)$$

(The exact punctuation depends on the logical system.)