Answer
a. $\forall$ people x, $\exists$ a person y such that x loves y.
b. negation: $\exists$ a person x, such that $\forall$ people y, x does not love y. In other words, there is someone who does not love anyone.
Work Step by Step
Recall the negation of a for all statement:
~(∀x in D, P(x)) $\equiv \exists x$ in D such that ~P(x).
Recall the negation of an exists statement:
~($\exists x$ in D, P(x)) $\equiv \forall x$ in D such that ~P(x).
To negate a multiply quantified statement, apply the laws in stages moving left to right along the sentence.
