Answer to Question #311852 in Discrete Mathematics for paul

Question

 Your answer sheets showing your name and solution.

1. Given the following:

·        g: "You can graduate."

·        m: "You owe money to the college."

·        r: "You have completed the requirements of your major."

·        b: "You have an overdue book."

Translate "You can graduate only if you have completed the requirements of your major, you do not owe money to the college, and you do not have an overdue book." into a propositional logic.

2. Show that  are logically equivalent.

3. Show, by the use of the truth table (truth matrix), that the  is a contradiction.

Answer 1

Solution (1)

Given that

·        g: "You can graduate."

·        m: "You owe money to the college."

·        r: "You have completed the requirements of your major."

·        b: "You have an overdue book."

We need to translate

"You can graduate only if you have completed the requirements of your major, you do not owe money to the college, and you do not have an overdue book."

You have completed the requirements of your major ... r

You do not owe money to the college ... ¬m

You do not have an overdue book ... ¬b

So, we need to put an AND operator between these conditions.

Therefore, we translate the above as

$g\iff (r\land \neg m \land \neg b)$

Solution (2)

The statement,

You can graduate only if you have completed the requirements of your major AND, you do not owe money to the college, AND you do not have an overdue book, which is logically equivalent.

The reason is that there is an "only if" condition, and that is possible only when all requirements are fulfilled or met.

Solution (3)

The required truth table is