CSCE 235

Homework Assignment 3

Assigned: January 30, 2008

Due: 12:30 p.m. February 8, 2008

(Homework 5 minutes late will not be accepted)

1. (5 points) Find a counterexample to prove that the following logical implication is false.

2. (40 points) For each of the following English statements, first translate it into symbolic notations using quantifiers and predicates, then negate it (and bring the negation inside the quantifiers), and then translate it back to English statements.

(a) (5 points) “Every computer scientist knows how to write a program.”

(b) (5 points) “Not all computer scientists can count.”

(c) (10 points) “There are some computer scientists who have been given the ACM Fellow Award.”

(d) (10 points) “Every computer scientist graduates from a university.”

(e) (10 points) “If a computer scientist is very good, he/she will be given the ACM Fellow Award.”

3. (10 points) Show that the premises “A car in this garage has an engine problem,” and “Every car in this garage has been sold” imply the conclusion “A car which has been sold has an engine problem.” Let be “x is in this garage,” be “x has an engine problem,” and be “x has been sold.” The premises are and . The conclusion is . Fill in the following blanks to complete the proof:

Step Explanation

1. _____________________

2. __________ Existential instantiation from (1)

3. _____________________

4. _____________________

5. _____________________

6. _______________________

7. __________ Simplification from (2)

8. __________ _______________________

9. __________ Existential generalization from (8)

QED