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February 22, 2013
Math 3345
Michael Tychonievich
Spring Semester 2013
Midterm 1
We will be going over some of the problems on Monday, 2/25. You may prepare one to talk about
for a few bonus points; please email me if you want to ensure a chance to speak.
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Autumn 2014
Math 3345 Syllabus
Instructor: Dr. William Husen
O ce: MW 124A
Phone: 292-5101
Email: husen@math.ohio-state.edu
WWW: http:/www.math.ohio-state.edu/husen
O ce Hours: 11:30-12:30 MWF (and by appointment).
Textbook: The Fundamentals of Higher Mat
Homework — Theory of Interest (Part 1)
1. Money accumulates in a fund at an effective annual interest rate of i during the ﬁrst
ﬁve years and at an effective annual interest rate of 2i thereafter.
A deposit of 1 is made into the fund at time 0. 1t accumul
Midterm # 2
1. Prove or Disprove. Let A, B, and C be nonempty sets and let f : A B and g : B C
both be bijections onto their respective targets. Let h = g f . Then h is a bijection from A
to C and h1 = f 1 g 1 .
2. Prove or Disprove. Let A, B, and C be se
April 24, 2013
Math 3345
Michael Tychonievich
Spring Semester 2013
Final Exam
Name: . . . . . . . . . . . . . . . . . . . . . .
Class time: . . . . . . . . . . . . . . . . . .
Instructions: Use the blank paper given to respond to exactly two problems in e
Mathematics 3345
Foundations of Higher Mathematics
Spring Semester 2016
Lecturer: Michael Tychonievich
Email: tychonievich.1@osu.edu
Office: MW629
Text: Mathematics 3345 Spring Semester 2016 by Neil Falkner. This book is
required for course readings and h
April 8, 2013
Math 3345
Michael Tychonievich
Spring Semester 2013
Practice Midterm 2
Respond to the following ve problems on your own paper. You should time yourself, giving
yourself 55 consecutive minutes to complete the entire practice test. Do not use
Name:
Math 3345 Practice Exam 2
October 18, 2016
SID:
I have read and understood the Code of Student Conduct, and this exam reflects my unwavering commitment to the principles of academic integrity and honesty expressed therein.
Signature:
Each part of ea
Name:
Math 3345 Practice Exam 1
September 9, 2016
SID:
I have read and understood the Code of Student Conduct, and this exam reflects my unwavering commitment to the principles of academic integrity and honesty expressed therein.
Signature:
Each part of e
Math 3345 Practice Exam 1 Answers
(1) (a) False. The negation is The Buckeyes sometimes lose.
(b) False. Its a tautology.
(c) True
(d) True
(e) False. Zero divides zero. (Work out why this is true!)
(2) (a) We did this in class. Look at your notes to find
Math 345
Equinumerous Intervals
The goal of these notes is to answer questions 5-9 in section 15 of the textbook. Rather than answer each one
individually, we will show that any non-degenerate interval in R is equinumerous to any other non-degenerate
inte
Math 3345
Section 4 - Unproven Results
The following are some results from section 4 that are necessary for many of the exercises in this section
but have yet to be proved. Most of them will be proved in section 7 on complete induction.
We have shown in c
Math 345
Contradictions and Proving Negatives
Proof by contradiction is a very powerful method of proof. Basically, if we wish to show that a given sentence
is always true, we assume that it is false and produce a contradiction. This means that our assump
Math 345
Converses, Contrapositives and Proof by the Contrapositive
The converse of the implication P ) Q is the reverse implication Q ) P . It is very important to realize
that these two implications are not logically equivalent.
Z
b
Example 1: From calc
Math 345
Truth Tables, Basic Equivalencies, Tautologies and Contradictions
Truth tables are not a primary focus in Math 345; however, it is important to know the truth tables of the
logical connectives. It is also important to understand how a truth table
Math 3345 Exam 1 Solutions
Problem 1. Suppose there are four cards lying on a table. One shows the number 4, one
shows the number 7, one is plain blue, and one is plain green. Which cards must you flip
over to determine if the following rule is valid?
A
Letian Yang
Math3345, 10:20
Instructor: C. Miller
Exercise: 10. 33(a),(b),(c),(e),(g)
10.33 We let A, B, C and D be sets
Proof 10.33(a): We prove that (A B)(C D)=(AC) (BD)
(x,y)(A B)(x,y)(C D)
iff (xA and yB) and (xC and yD)
iff (xA and xC) and (xB and yD
Letian Yang
Math3345, 10:20
Instructor: C. Miller
Exercise: 10. 5, 8, 10, 11
10.5 We let S be a set such that for each set A, we have S A. We show that S=.
Recall Proposition 10.6: For each set A, we have A
Proposition 10.5 (b): For all sets A and B, if A
Letian Yang
Math3345, 10:20
Instructor: C. Miller
Exercise: 4.1 (a) (b), 2 (a), 12, 13, 14
4.1 (a) We prove that if x is even and y is even, then x+y is even.
(b) We prove that if x is even and y is odd, then x+y is odd.
Proof (a).
Suppose x is even and y
Letian Yang
Math3345, 10:20
Instructor: C. Miller
Exercise: 3.10 (a)-(d), 14 (b)-(h)
3.10 For each of the following sentences, we write out what it means in words. We state whether it is
true or false and we prove statement.
Proof (a).
The sentence (yR) (
Letian Yang
Math3345, 10:20
Instructor: C. Miller
Exercise: 3.1 (e)-(k); 3.5
3.1 For each of the following sentences, we write out what it means in words. We state
whether it is true or false and prove statement.
Proof (e).
The sentence (xR)(x2-4x+3>0) me
Letian Yang
Math3345, 10:20
Instructor: C. Miller
Exercise: 7.4, 5
7.4 (a) We suppose S is a subset of N and S does not have a least element. Show that S is
empty.
Proof 7.4(a): Suppose that the complete induction holds. We establish the contrapositive of
Letian Yang
Math3345, 10:20
Instructor: C. Miller
Exercise: 5.3 (a), 4, 5, 6
5.3 (a) We prove by induction that for each nN, 13+23+n3=(n2(n+1)2)/4
Proof 5.3(a): Let P(n) be 13+23+n3=(n2(n+1)2)/4. For P(1), the formula holds for true for
n=1, 13= (12(1+1)2