Assignment 1, due at the beginning of class, Wednesday, May 12
n
i2 =
1. Prove that
i=1
n(n + 1)(2n + 1)
for all n 1.
6
n
i2 =
Solution: Let P (n) denote the predicate
i=1
n(n + 1)(2n + 1)
, and let n0 = 1. We rst prove that
6
1
1(1 + 1)(2(1) + 1)
(1)(2)
Students Name [print]
Student Number
Mathematics 2155a Second Midterm Exam
9:30amnoon
November 1, 2008
Instructions: Print your name on the SCANTRON answer sheet. Sign the SCANTRON answer sheet,
and mark your student number on the SCANTRON answer sheet. U
Math 2155 Assignment 3: due at beginning of class on Monday, October 20
1. Prove or disprove the following assertion: For any set A, and any relations R, S on A,
(R S)1 = R1 S 1 .
Solution:
(R S)1 = cfw_ (a, b) A A | (b, a) R S
= cfw_ (a, b) A A | (b, a)
Math 2155 Assignment 5: due at beginning of class on Monday, November 17
1. Let B = cfw_ 1, 2, 3, 4, 6, 8, 12, 16 . Draw the Hasse diagram for (B, D), where D is the
restriction to B of the division ordering on Z+ ; that is, for b1 , b2 B, b1 D b2 if and
Math 2155 Assignment 4: due at beginning of class on Wednesday, October 29
1. Let S be a set, and let A = P(S). Choose F S, and let R A A denote the relation
on A dened by
R = cfw_ (X, Y ) A A | X F = Y F .
(a) Prove that R is an equivalence relation on A
Assignment 1, due at classtime, Wednesday, September 22
1. Given that the following is true: for all n 3, 2n + 1 < 2n , prove that
n 3 (n2 < 2n ) (n + 1)2 < 2n+1 )
is true.
Solution: Let n 3. If n2 2n , then the implication (n2 < 2n ) (n + 1)2 < 2n+1 ) is
Assignment 2, due at classtime, Wednesday, September 29
1. Suppose that instead of having the Well Ordering principle as a given fact about the
natural numbers, we were given the principle of mathematical induction as a fact. In this
problem, we are to be
Students Name [print]
Student Number
Mathematics 2155A First Midterm Exam
October 15, 2010
6:30am9:00pm
Instructions: Print your name and your instructors name on the SCANTRON answer sheet. Sign the
SCANTRON answer sheet, and mark your student number and
Assignment 6, due at the beginning of class, Wednesday, October 27
n
i
n
1. Prove that for any integers m and n with 0 m n, i=m n m = 2nm m .
i
One way would be to prove it by induction. If one wanted to use n as the induction variable, then a
n
i
n
suita
Assignment 5, due at my oce, MC 132, by 5:00pm, Friday, October, October 22
1. (a) How many arrangements of the letters of the word SEQUENCES have at least two
letters between the Q and the U?
Solution: We reserve two blanks on which to place the Q and U,
Assignment 4, due at classtime, Wednesday, October 13
1. Let n be a positive integer.
(a) How many binary sequences of length n are there? That is, what is the size of the set
cfw_ 0, 1 n?
Solution: |cfw_ 0, 1 n| = |cfw_ 0, 1 |n = 2n .
(b) How many binary
Assignment 3, due at classtime, Wednesday, October 6
1. Prove that for any sets A, B, C, if (A B) C = (A C) B, then B C = .
Solution: Let A, B, C be sets, and suppose that (A B) C = (A C) B. We must then prove that
B C = . Suppose to the contrary that B C
Students Name [print]
Student Number
Mathematics 2155A Second Midterm Exam
6:30am9:00pm
November 12, 2010
Instructions: Sign the SCANTRON answer sheet, and mark your student number on the
SCANTRON answer sheet. Use a PENCIL to mark your answers to questio
Assignment 9, due at the beginning of class, Wednesday, November 24
If A is a nite set on which we have a partial order relation R, then we dene the height
of (A, R), denoted by h(A,R) , or often just by hA when there is no doubt as to which
relation R is
Math 2155 Assignment 2: due at beginning of class on Wednesday, October 1
1. Prove that for any sets A, B, and C, (A B) C = (A C) B if and only if B C = .
Solution: There are many ways to tackle this problem.
Solution 1. Suppose that B C = , and let x B C
Students Name [print]
Student Number
Mathematics 2155a First Midterm Exam
9:30amnoon
October 4, 2008
Instructions: Print your name and your instructors name on the SCANTRON answer sheet. Sign the
SCANTRON answer sheet, and mark your student number and sec
Assignment 2, due at the beginning of class, Thursday, May 13
1. Prove that for all sets A, B, and C, if (A B) (A C) = A (B C), then
A B A C.
Solution: Let A, B, and C be sets such that (A B) (A C) = A (B C). We are to prove that
A B A C. Let x A B. Then
Students Name [print]
Student Number
Mathematics 2155A Intersession Final Exam
2:005:00 p.m.
May 29, 2010
Instructions: Print your name on the SCANTRON answer sheet. Sign the answer
sheet, write your student number on the SCANTRON answer sheet. Use a PENC
Assignment 7, due at the beginning of class, Friday, May 21
1. (a) How many functions are there from a set of size 6 to a set of size 3?
Solution: There are 36 = 93 = 729 functions from a set of size 6 to a set of size 3.
(b) Evaluate S(6, 3), the number
Math 2155 Assignment 1: due at beginning of class on Monday, September 22
1. Prove that xn y n = (x y)
n1 i n1i
i=0 x y
for all n 1 (by denition, x0 = y 0 = 1).
Solution: Here is one way to do it (without induction). Let n 1. Then
n1
n1
n1
xi y n1i
xi y n
Students Name [print]
Student Number
Mathematics 2155A Midterm Exam
7:009:30 p.m.
19 May, 2010
Instructions: Print your name and your instructors name on the SCANTRON
answer sheet. Sign the answer sheet, and mark your student number and section on the
SCA
Assignment 3, due at the beginning of class, Friday, May 14
1. A 4-digit number is a sequence of 4 base 10 digits in which the rst digit is not 0. The
4-digit integer d1 d2 d3 d4 is equal to d4 + d3 (10) + d2 (102 ) + d1 (103 ).
(a) How many 4-digit numbe
Assignment 4, due at the beginning of class, Monday, May 17
1. (a) How many arrangements of the letters of the word MISSISSIPPI have no two Ss
adjacent (side-by-side)?
Solution: We partition the set of all such arrangements according the the 4-tuple of in
Assignment 5, due at the beginning of class, Tuesday, May 18
1. (a) Give an example of symmetric relations R and S on J3 such that R S is not symmetric.
Provide a proof that your relations do indeed have the required property.
Solution: Let S = cfw_ (1, 2
Assignment 6, due at the beginning of class, Wednesday, May 19
1. (a) Let A be a set and R A A. Prove that if R+ is antisymmetric, then R is antisymmetric.
Solution: Since R R+ and so R1 (R+ )1 , we have R R1 R+ (R+ )1 1A . Thus R is
antisymmetric.
(b) Gi
Assignment 11, due at the beginning of class, Friday, May 28
Let H = cfw_ 1J4 , 1 2 3 4 , 1 2 3 4 , 1 2 3 4 , , so H is a subset of the symmetric group S4 .
4321
3412
2143
In fact, H is a subgroup of S4 , but you are not asked to prove this in its entiret
Assignment 8, due at the beginning of class, Tuesday, May 25
1. Let A and B be sets, and let f : A B be a function. Recall that for any subset X
of A, we dened f (X) = cfw_ f (x) | x X . As well, for any subset Y of B, we dened
f 1 (Y ) = cfw_ a A | f (a)
Assignment 9, due at the beginning of class, Wednesday, May 26
Recall that we have dened Z2 to be the set of all congruence classes of Z modulo 2, so
Z2 = cfw_ [0]2 , [1]2 . For convenience of writing, we shall just write i for [i], but one must
then reme
Assignment 10, due at class time, Thursday, May 27
1. Dene a binary operation on A = R cfw_ 1, 1 by
(x, i) (y, j) = (y + xj, ij)
for all x, y R, i, j cfw_ 1, 1 (note that i, j are each either 1 or 1, and so ij is either 1
or 1).
(a) Prove that is associ
Assignment 7, due at the beginning of class, Wednesday, November 3
1. Let A, B, C be sets, let R be a relation from A to B, and let S and T be relations from B
to C, so that S T is a relation from B to C. Prove that (S T ) R = (S R) (T R).
Solution: Since