Answer
a. $\forall xL(x, Jerry)$
b. $\forall x \exists y L(x, y)$
c. $\exists y \forall xL(x, y)$
d. $\neg \exists x \forall y L(x,y)$
e. $\exists y \neg L(Lydia, y)$
f.$\exists y \forall x \neg L(x, y)$
g.$\exists x(\forall y L(x, y) \land \forall z((\forall wL(w, z)) \rightarrow z = x))$
h.$\exists y \exists z(L(Lynn, y) \land L(Lynn, z) \forall y \ne z \land \forall w(L( Lynn, w) \rightarrow (w=y \lor w = z))$
(i) $\forall xL(x, x)$
(j) $\exists x \forall y(L(x,y) \leftrightarrow x = y)$
Work Step by Step
a. "Everybody” means ”All people in the world”.
$\forall xL(x, Jerry)$
b. ’Everybody” means "All people in the world”. ’’Somebody” means ’’There exists a person in the world”
$\forall x \exists y L(x, y)$
c. ’’Somebody” means "There exists a person in the world”. ’’Everybody” means " All people in the world”.
$\exists y \forall xL(x, y)$
d. ’’Nobody" means ’’There does not exists a person in the world”. ’’Everybody" means “All people in the world”.
$\neg \exists x \forall y L(x,y)$
e. We could rewrite the given sentence as "Lydia does not love somebody". "Somebody" means “There exists a person in the world"
$\exists y \neg L(Lydia, y)$
f. We could rewrite the given sentence as ’’There is somebody whom everybody does not love". (Note: Lydia refers to x and somebody refers to y. while the statement of y occurs before the statement of x)
$\exists y \forall x \neg L(x, y)$
g. We could rewrite the given sentence as "There is somebody x whom everyone loves and all people that are loved by everyone, then this person has to be x".
$\exists x(\forall y L(x, y) \land \forall z((\forall wL(w, z)) \rightarrow z = x))$
h. We could rewrite the given sentence as "There are two people y and z that Lynn loves and these two people are different and for all people that Lynn loves, these people then have to be either y or z".
$\exists y \exists z(L(Lynn, y) \land L(Lynn, z) \forall y \ne z \land \forall w(L( Lynn, w) \rightarrow (w=y \lor w = z))$
(i) We could rewrite the given sentence as "Every person x loves x (himself/herself)"
$\forall xL(x, x)$
(j) We could rewrite the given sentence as "There is a person x who loves y if and only if y is x (himself/herself)".
$\exists x \forall y(L(x,y) \leftrightarrow x = y)$
