1. Cardinality of Sets
In this section, we pursue the notion of the size of a set. Recall that a set X is said to be nite
if there exists a natural number n and a bijective function from X to Jn (where Jn = cfw_ 1, 2, . . . , n
if n 1, and J0 = ), in whi
Math 2155F Assignment 4, due at classtime, Wednesday, October 15
Please leave some space between your answers in order to provide us with room for comments.
1. Decide whether or not for all sets A, B, and C, (A B) C) (A C) = A (B C).
Give a proof in suppo
Math 2155F Assignment 3, due at classtime, Monday, October 6
1. Let an , n = 1, 2, . . . denote the innite sequence dened inductively by the requirements
1
a1 = 1 and an+1 = 3
for all integers n 1.
an
(a) Prove that an < an+1 < 3 for every integer n 1.
S
Math 2155F Assignment 1, due at classtime, Monday, September 22
1. A set A is said to be transitive if for all x A, x A.
(a) Prove that every natural number is transitive.
Solution: Once you see what this entails, it follows immediately from two results i
Name:
Student Number:
Mathematics 2155A Midterm Exam
October 25, 2014
1:004:00pm
1. In each case below, answer true or false (no reasons need to be provided).
2
marks
(a) cfw_ .
2
marks
(b) cfw_ .
2
marks
(c) cfw_ cfw_ .
2
marks
(d) For all sets A, B, an
Math 2155F Assignment 1, due at classtime, Monday, September 22
1. Prove that the sum of any ve consecutive integers is a multiple of 5.
Solution: Given any set of ve consecutive integers, let n denote the smallest of these integers, so the
4
4
integers a
Math 2155F Assignment 2, due at classtime, Friday, October 2
1. Prove that for all sets A, B, and C, A (B C) = (A B) (A C).
2. Prove that for every n N, if n = 0, then there exists a unique m N such that n = m+ .
3. A set A is said to be transitive if for
Assignment 1, due at classtime, Wednesday, September 30
1. Let a1 = 1 and for n 1, let an+1 = 3
1
. Use mathematical induction to prove that
an
an an+1 for each integer n 1.
Solution: First attempt: let P (n) denote the predicate an an+1 , and n0 = 1. In
Assignment 5, due at classtime, Wednesday, October 28
1. How many binary strings of length 12 (that is; sequences of length 12 whose entries are
chosen from cfw_ 0, 1 ) have every occurrence of two 1s separated by at least two 0s? For
example, 10010001000
Math 2155F Assignment 5, due at classtime, Monday, October 20
Please leave some space between your answers in order to provide us with room for comments.
1. Suppose that an , n 1, is a sequence of real numbers for which it is known that an an+1
for every
Math 2155F Assignment 6, due at classtime, Monday, November 3
Please leave some space between your answers in
order to provide us with room for comments.
1. (a) How many arrangments are there of the letters in the word SENSELESSNESS?
Solution: There are S
Students Name [print]
Student Number
Mathematics 2155a Final Exam
7:0010:00pm
December 13, 2013
Instructions: Print your name on the SCANTRON answer sheet. Sign the SCANTRON
answer sheet, and use a PENCIL to record your student number on the SCANTRON
answ
Mathematics 2155a
Final Exam
December 13, 2013
Instructions: Print your name on the SCANTRON answer sheet. Sign the SCANTRON
answer sheet, and use a PENCIL to record your student number on the SCANTRON
answer sheet. Use a PENCIL to mark your answers to qu
Math 2155F Assignment 11, due by 5:00pm Friday, December 5 (at my oce)
1. Let A be a nonempty set, and let be a binary operation on A. For each a A, let
a : A A denote the function for which a (x) = a x for all x A.
(a) Prove that if is associative, then
Math 2155F Assignment 10, due at classtime, Monday, December 1
Please leave some space between your answers in
order to provide us with room for comments.
1. (a) Let A be a set. Prove that if f : A A satises f f = 1A , then
P = cfw_ cfw_ x, f (x) | x A ,
Math 2155F Assignment 9, due at classtime, Monday, November 24
Please leave some space between your answers in
order to provide us with room for comments.
1. Let A be a set, and let B be a subset of A. Dene a relation R on P(A) by
R = cfw_ (X, Y ) P(A) P(
Math 2155F Assignment 8, due at classtime, Monday, November 17
Please leave some space between your answers in
order to provide us with room for comments.
1. Let A, B, and C be sets, and let R and S be relations from A to B, and let T be a relation
from B
Math 2155F
Assignment #2
Raj Pathak
1. Prove that for all sets A, B, and C, A (B C) = (A B) (A C).
Let A, B and C be sets. A (B C) can be rewritten as (A A) (B C), since the
union of sets is idempotent. When we apply the Cartesian product of (A A) (B C),
Math 2155F Assignment 7, due at classtime, Monday, November 10
Please leave some space between your answers in
order to provide us with room for comments.
1. Let A be a set, and let R be a relation on A. Prove that for all positive integers m and n,
(Rm )
Assignment 4, due at classtime, Wednesday, October 21
1. How many binary strings of length 7 (that is; sequences of length 7 whose entries are
chosen from cfw_ 0, 1 ) contain the substring 101? For example, 1010101 is such a string,
while 0111110 is not.
Assignment 3, due at classtime, Wednesday, October 14
1. Prove that for any sets A, B , and C , if (A B ) C = (A C ) B , then B C = .
Solution: We shall use the contrapositive. Suppose that A, B , and C are sets such that B C = . Then
we must prove that (
Assignment 2, due at classtime, Wednesday, October 7
1. Use mathematical induction to prove that n < Fn for every integer n 6, where Fn is the
nth Fibonacci number (F0 = 0, F1 = 1, and Fn+2 = Fn+1 + Fn for n 0).
Solution: Let P (n) denote the assertion th
Mathematics 2155A Assignment 6
Due at the beginning of class, Wednesday, October 23
1. A 4-digit integer is a sequence of length 4 with entries from cfw_ 0, 1, 2, . . . , 9 whose rst
entry is not 0. A 4-digit integer is said to be odd if its fourth entry
Mathematics 2155A Assignment 5
Due at the beginning of class, Wednesday, October 16
1. Prove that for any sets A and B , (A B ) (A B ) = A B .
Solution: Let A, B , and C be sets. Let x (A B ) (A B ). Then either x A B and x A B ,
/
or else x A B and x A B
Mathematics 2155A Assignment 9
Due at the beginning of class, Wednesday, November 20
1. Let A be a set, and suppose that f : A A satises f 3 = 1A . Prove that f is bijective.
Solution: f 2 is both a left and a right inverse of f , so f is injective and su
Mathematics 2155A Assignment 10
Due at the beginning of class, Wednesday, November 27
1. Let (A, R) and (B, S ) be partially ordered sets. A function f : (A, R) (B, S ) (that
is, f : A B , and we are told that R is the partial order relation on the domain
Students Name [print]
Student Number
Mathematics 2155a Midterm Exam
6:309:30pm
October 26, 2012
Instructions: Print your name on the SCANTRON answer sheet. Sign the SCANTRON answer sheet,
and mark your student number and section on the SCANTRON answer she
Students Name [print]
Student Number
Mathematics 2155a Midterm Exam
6:309:30pm
October 25, 2013
Instructions: Print your name and your instructors name on the SCANTRON answer
sheet. Sign the SCANTRON answer sheet, and mark your student number and section
Mathematics 2155a
Midterm Exam
October 26, 2012
Instructions: Print your name on the SCANTRON answer sheet. Sign the SCANTRON
answer sheet, and mark your student number and section on the SCANTRON answer
sheet. Use a PENCIL to mark your answers to questio
Mathematics 2155A Assignment 11
Due at the beginning of class, Wednesday, December 4
1. (a) Prove that in a nite monoid M , the product of two non-invertible elements is noninvertible. Hint: use contradiction, and consider for any m M the function left
mu