CSCE 235

Homework Assignment 3

Assigned: January 28, 2009

Due: 12:30 p.m. February 4, 2009

(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 are smart.”

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

(d) (10 points) “For all computer scientists, if the computer scientist is very good, then understanding logic well is necessary.”

(e) (10 points) “Every computer scientist has at least a friend who is also a computer scientist.”

3. (15 points) Show that the hypotheses “Some animals in this zoo Z have a name,” and “Every animal in Z has its own enclosure in zoo Z” imply the conclusion “There is at least one animal in Z that has a name and also its own enclosure in Z” Suppose x is the universe of discourse for “animals”. Let be “x is in zoo Z,” be “x has a name”, and be “x has its own enclosure in Z”. What should the premises/hypotheses be? What is the conclusion that you need to prove?