CSCE 235
Homework
Assignment 3 - Solution
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.
Suppose that
a is the only element in the universe of
discourse x. Suppose that 
is true but 
is false. Now, we can see that the left hand side is
true since there is one element in x that 
is true (i.e., a). However, the right hand side is false. We cannot find an element in x that
satisfies both 
and 
! So, the logical
implication is false. The counterexample is thus: 
is true and is false, where a is the only element of the universe of discourse x.
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.
First, let us suppose that x
is the universe of discourse for “computer scientists”.
(a) (5 points) “Every computer scientist knows how to write
a program.”
The translation is: 
The negation is: 
The translation of the negation
is: “There is at least a computer
scientist who does not know how to write a program.”
(b) (5 points)
“Not all computer scientists are smart.”
The translation is:
The negation is:
The translation of the negation
is: “All computer scientists are smart.”
(c) (10 points) “There are some computer scientists who have
been given the Turing Award.”
The translation is:
The negation is: 
The translation of the negation
is: “Every computer scientist has not
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.”
The translation is:
The negation is: 
implication
De Morgan
The translation of the negation
is: “There is at least a computer
scientist who is very good but does not understand logic well.”
(e) (10 points) “Every computer scientist has at least a friend who is also a computer scientist.”
Here, let us suppose that y
is the universe of discourse for “humans”. (If you don’t have this, and only
work from “x is the universe of discourse for computer scientists, your
solution is likely to be very cumbersome.)
The translation is:
The negation is: 
De Morgan
Implication
The translation of the negation
is: “There is at least one computer
scientist such that if somebody is his/her friend, then that somebody is not 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?
The premises are 
and 
. The conclusion is
.
Step Explanation
1.
Hypothesis
2.
Hypothesis
3.
1;
Existential Instantiation
4.
2;
Universal Instantiation
5.
3; Simplification
6. 
4,
5; Modus Ponens
7.
3, 6; conjunction
8.
7; Existential Generalization
Q.E.D.