Answer
$((r \land \lnot m \land \lnot b) \leftrightarrow g).$
Work Step by Step
Are propositions are
g: "You can graduate,"
m: "You owe money to the university,"
r: "You have completed the requirements of your major," and
b: "You have an overdue library book."
We want to translate the statement "You can graduate only if you have completed the requirements of your major and you do not owe money to the university and you do not have an overdue library book" into propositional logic.
First, we want to capture the meaning of the sentence.
Note that as in the previous problem, there is an implicit biconditional connective; the 'only if' is implicitly an 'if and only if'. Because what this statement means is in order to graduate you must meet three conditions; you must complete the requirements for your major, you must not owe the university money, and you must not have an overdue library book, and if you do not meet any one of these three conditions, you will not graduate.
Next, we rephrase our statement to emphasize its meaning using logical connectives. We rephrase it as the two conditional statements "IF you complete the requirements of your major AND it is not the case that you owe money to the university AND it is not the case that you have an overdue library book, THEN you can graduate." and "IF it is not the case that (you complete the requirements of your major AND it is not the case that you owe money to the university AND it is not the case that you have an overdue library book), THEN it is not the case that you can graduate."
Hence, an appropriate translation is $((r \land \lnot m \land \lnot b) \rightarrow g) \land (\lnot (r \land \lnot m \land \lnot b) \rightarrow \lnot g)$
Recall that a conditional statement is equivalent to its contrapositive, so $\lnot (r \land \lnot m \land \lnot b) \rightarrow \lnot g$ is equivalent to $g \rightarrow (r \land \lnot m \land \lnot b)$.
Thus our translation is equivalent to $((r \land \lnot m \land \lnot b) \rightarrow g) \land (g \rightarrow (r \land \lnot m \land \lnot b)),$ which is equivalent to $((r \land \lnot m \land \lnot b) \leftrightarrow g).$
