Math 220 - 902
Final Exam Solutions
May 11, 2005
1. (20) Write the negation of the following statement:
n N (m N ) (p N ) such that mn np N .
The negation of this statement is
n N such that (m N ) (p N ) mn np N
2. (20)
(a) Dene an equivalence relation on
Math 220 - 901
1.
Final Exam Solutions
(15) Let A be a set and R a relation on A. That is, R A A.
a. Define what it means to say that R is an equivalence relation.
R is an equivalence relation if it satisfies the following three conditions:
reflexive: a A
Math 220 - 502
Solutions Exam 2
November 5, 2004
1. (30) Let f : N N , where N is the set of natural numbers cfw_1, 2, . Suppose that
f (n) = n2 + 1.
(a) If f injective (one-to-one)?
Yes f is injective. Suppose f (n) = f (m). This implies
m2 + 1 = n 2 + 1
Math 220 - 901
1.
(15) Define the following terms:
a. f : X
Y,
This denotes a function from X to Y. That is a subset of X
conditions are true:
f
for all x X there is a y Y such that x, y
if x, y 1 and x, y 2 are in f then y 1 y 2 .
b.
c.
2.
Exam 2 Solutio
Math 220 - 502
Solutions Exam 1
October 1, 2004
1. (15) Dene the following terms:
(a) If A and B are two sets, what does A B mean?
A B means that A is a subset of B . That is, if x A, then x B .
(b) If A and B are two sets what is A B ?
A B = cfw_(a, b) :
Math 220 - 901
1.
Exam 1 Solutions
February 25, 2008
(10) Write a statement that is logically equivalent to the conditional statement P Q, using only "and
", "or
" and "negation". Be sure to prove that the two statements are logically equivalent.
The foll
Math 220 - 502
Final Exam Solutions
May 12, 2004
1. (20) Let A, B , and C denote three sets. Prove or disprove the following:
(a) (A B ) C = (A C ) (B C )
This is a true statement. To see this Ill show that each side of the equality is a
subset of the oth
May 7, 1997
Math 220
Final Exam
A yes or no answer or an answer with no justi cation will not be acceptable. Remember to write
neatly, clearly, and in sentences.
1. 15 If n is an even integer, show that gcdn; n + 2 = 2. What can you say about the
gcdn; n
Math 220 - 502
Exam 2 Solutions
March 31, 2004
1. (15) Dene the following:
(a) a transitive relation on a set A. Note: there are two terms that need to be dened. They
are relation and transitive.
First a relation on a set A is just a subset of A A the Car
Math 220
Solutions Exam 2
April 1, 2005
1. (20) Dene the following terms:
(a) f is a function from the set A to the set B ,
A function from a set A to a set B is a subset of A B such that
a A, b B such that (a, b) f
a A, (a, b1 ) and (a, b2 ) in f implies
April 3, 1997
Math 220
Exam 2
A yes or no answer or an answer with no justi cation will not be acceptable. Remember to write
neatly, clearly, and in sentences.
1. 15 De ne the following terms, and give an example of each.
a Relation in A B ,
Ans: A relati
Math 220 - 502
Exam 1 Solutions
February 25, 2004
1. (30) Dene the following:
(a) If A and B are two sets, what is A B ?
A B = cfw_x : x A and x B
/
(b) If A and B are two sets, what is A B ?
A B = cfw_x : x A or x B
(c) If f : A B , and X A, what is f
Math 220 - 902
Solutions to Exam 1
February 18, 2005
1. (20) Dene the following terms:
(a) Statement (in the context of logic)
A statement is a declarative sentence, which is either true or false.
(b) The Cartesian product of two sets A and B .
The Cartes
Feb. 20, 1997
Math 220
Exam 1
A yes or no answer or an answer with no justi cation will not be acceptable. Remember to write
neatly, clearly, and in sentences.
1. 20 De ne the following terms and or symbols, and give an example of each. No example,
no cre