Do the following two problems below (they are from the text) and email me your answers:

• Problem 8.9 (page 316) parts a through d. You can try part e for extra credit.
  
  
• Problem 8.10 (page 317) parts a through c.

Problem 8.9:

This exercise uses the function MapColor (x) and predicates In (x,y), Borders (x,y), and Country (x), whose arguments are geographical regions, along with constant symbols for various regions. In each of the following we give an English sentence and a number of candidate logical expressions. For each of the following expressions, state whether it

1. correctly expresses the English sentence;
2. is syntactically invalid and therefore meaningless; or
3. is syntactically valid but does not express the meaning of the English sentence.

Consider a vocabulary with the following symbols:

Occupation(p,o): A predicate that means that Person p has occupation o.

Customer(p1,p2): A predicate that means that Person p1 is a customer of Person p2.

Boss(p1,p2): A predicate that means that Person p1 is a boss of Person p2.

Doctor, Surgeon, Teacher, Lawyer, Actor: Constants that denote occupations.

Emily, Joe: Constants that denote Persons.

Use these symbols to write the assertions in parts a through g in first order logic. Here are a few examples:

Occupation(Paul,Teacher) means "Paul is a Teacher."

Boss(Dino,Paul) means "Dino is a boss of Paul."

Occupation(Bob,Lawyer) ∧ Occupation(Bob,Teacher) means "Bob is a Lawyer and a Teacher."

∀x Occupation(x,Teacher) ⇒ Occupation(x,Actor) means "Every teacher is an actor."

Now the homework:

a. Emily is either a surgeon or a lawyer.

b. Joe is an actor, but he also holds another job.

c. All surgeons are doctors.

d. Joe does not have a lawyer (meaning that he is not a customer of any lawyer.)

e. Emily has a boss who is a lawyer.

f. There exists a lawyer all of whose customers are doctors.

g. Every surgeon has a lawyer.