FIRST EXAMSTUDY COPY
MATH 251, FALL 2006, WILLIAMS COLLEGE
OCTOBER 12, 2006
These are the problems from the rst midterm exam.
1. Problem One
Show that the following argument is valid.
p
q p
q
2. Problem Two
(1) Give an example of a relation which is both
Final Exam
Math 251, Williams College
December 16, 2006
Directions: This Exam has 6 questions on 8 pages. Be
sure to show all of your work and explain yourself carefully.
You have two and one half hours to complete this exam.
Please sign the honor code pl
HOMEWORK ASSIGNMENT # 1
MATH 251, FALL 2006, WILLIAMS COLLEGE
1. Problem One
Construct the truth table of q (p) (q).
1.1. Solution. To t the table on the page, we use x to denote
(p) (q).
p
T
T
F
F
q p q (p) (q) = x q x x q q x
T F F
F
F
T
F
F F T
T
T
F
F
SECOND EXAM SOLUTIONS
MATH 251, WILLIAMS COLLEGE, FALL 2006
Abstract. These are the instructors solutions. For statements of the problems, see the posted exam. You should take these more as a quick guide on
how to do the problems than as a representative
HOMEWORK ASSIGNMENT # 2
MATH 251, FALL 2006, WILLIAMS COLLEGE
Abstract. These are the instructor solutions to homework 2.
1. Problem One
Using Venn diagrams, investigate whether the following statements
are true or false.
(1)
(2)
(3)
(4)
(5)
A (B C) = (A
HOMEWORK ASSIGNMENT # 3 SOLUTIONS
MATH 251, FALL 2006, WILLIAMS COLLEGE
Abstract. These are the instructors solutions.
1. Problem One: On injections
Let f : A B and g : B C be functions.
(1) Prove that if f and g are injective, then g f is injective.
(2)
HOMEWORK ASSIGNMENT 4
MATH 251, WILLIAMS COLLEGE, FALL 2006
Abstract. These are the instructors solutions.
1. A theorem on sets
Prove that for any sets A, B1 , . . . , Bn ,
n
n
A
(1)
Bi
(A Bi ).
=
i=1
i=1
1.1. Solution. We proceed by induction on the numb
HOMEWORK ASSIGNMENT 5
MATH 251, WILLIAMS COLLEGE, FALL 2006
Abstract. These are the instructors solutions.
1. Big Brother
The social security number of a person is a sequence of nine digits that
are not necessarily distinct. If X is the set of all social
MATH 251 HOMEWORK 7 SOLUTIONS
FALL 2006, WILLIAMS COLLEGE
Abstract. These are the instructors solutions.
1. The Towers of Hanoi
Three vertical cylindrical poles of equal radius and height are place along a line
on top of a table and n circular disks of de
MATH 251 HOMEWORK 6
FALL 2006, WILLIAMS COLLEGE
Abstract. These are the instructors solutions.
1. Collections
Find the number of r-collections that can be formed using the elements of the
set X = cfw_A, B, C, D, E, F, G if
r = 4 and the elements are dist
Second Exam
Math 251, Williams College
November 9, 2006
Directions: This Exam has 7 questions. Be sure to show all of your work
and explain yourself carefully. You have three hours to complete this exam,
to be done in a single block of time. You may not u