Please explain the logic behind the truth values of these expressions

Can someone explain the reasoning behind the answers given? We reviewed these in class, but I wasn't able to grasp the logic. Especially c, d, and e.

The domain of possible values for variables X and Y is {Jim, Ann, Sal, Pat, Tom}. The following facts define the values for which the child predicate is true. The child predicate is false for all other cases.

Evaluate each expression using the domain values and predicates as defined and indicate if the expression is true or false.

child (Ann,Jim)

child (Sal,Jim)

child (Pat,Ann)

child (Tom,Sal)

a. $(\forall X)$ child$(X,Jim)$

b. $(\exists X)\neg$child$(X,Jim)$

c. $(\forall X)(\exists Y)($child$(X,Jim) \rightarrow$child$(Y,X))$

d. $(\exists Y)(\forall X)($child$(X,Jim) \rightarrow$child$(Y,X))$

e. $(\exists X)(\forall Y)($child$(X,Jim) \rightarrow$child$(Y,X))$

a. FALSE (Reasoning: Any case that make it false? Yes, so FALSE)

b. TRUE (Reasoning: Any case that makes it true? Yes, so TRUE)

c. TRUE

d. FALSE

e. TRUE

Let me interpret $child(x,y)$ as "$x$ is a child of $y$", then:
(E)The interpretation of this one would be "there is someone who verifies that if she/he is a child of Jim then everybody is a child of her/him", if yoy substitute Jim for your variable $x$ then the implication is true because the antecedent (Jim is a child of Jim) is false, so you found someone who verifies that, and so it is true.