BAIN MUSC 525 Post-Tonal Music Theory
Set Theory
"A set may be viewed as any well-defined collection of objects; the objects are called the elements or members of the set." "The concept of a set appears in all branches of mathematics. This concept f
Math 220
Exam 2 Partial Solutions
March 30, 2012
S. Witherspoon
n 2,
2n,
1. Let f : Z Z be dened by f (n) =
if n is even
if n is odd
(a) [5 points] Find f (cfw_1, 2, 3, 4).
Since 1 and 3 are odd, f (1) = 2 and f (3) = 6. Since 2 and 4 are even, f (2) =
2
Math 220
Exam 1 Partial Solutions
September 2012
S. Witherspoon
1. (a) For all integers m and n, if m is even or n is even, then mn is even.
(b) For all integers m and n, if m is odd and n is odd, then mn is odd.
(c) There exist integers m and n such that
Math 220
Exam 2 Partial Solutions
November 9, 2012
S. Witherspoon
1. (a) f (O) = E, the set of even integers.
(b) f 1 (E) = Z since the image of each integer, under f , is even.
(c) No: For example, f (2) = f (3) but 2 = 3.
2. (a) There are many possible
Math 220 Exam 1
February 16, 2007
S. Witherspoon
Name
There are 5 questions, for a total of 100 points. Point values are written beside each
question.
1. Consider the following proposition: For all real numbers x, if x > 2, then x2 > 4.
(a) [5 points] Wri
Math 220 Exam 2
March 28, 2007
S. Witherspoon
Name
There are 5 questions, for a total of 100 points, plus one bonus question worth 5
points. Point values are written beside each question.
1. Let R = cfw_(1, 2), (2, 3), (2, 2), (3, 2), (4, 3).
(a) [6 point
Math 220 Exam 1
February 15, 2012
S. Witherspoon
Name
There are 5 questions, for a total of 100 points. Point values are written beside each
question.
1. Consider the statement: For all integers m and n, if m is odd, then mn is odd.
(a) [5 points] Write t
Math 220 Exam 2
March 30, 2012
S. Witherspoon
Name
There are 6 questions, for a total of 100 points. Point values are written beside each
question.
n 2,
2n,
1. Let f : Z Z be dened by f (n) =
if n is even
if n is odd
(a) [5 points] Find f (cfw_1, 2, 3, 4)
MATH 220 Writing Assignments
Fall 2012, Section 905
There are three writing assignments in addition to the writing you will do for
homework and exams: Two are extended solutions to discussion and discovery exercises from the text (minimum 500 words each,
MATH 220-970
Last Homework Solutions
FALL, 2012
Section 5.4
6c. Suppose n 2 is rational. Then there exist a, b Z, gcd(a, b) = 1 such that n 2 = a/b.
Thus 2 = an /bn , whence 2bn = an . Since 2|2bn , 2|an , so 2|a. Thus there exists an
integer c so that a
MATH 220-901
Last Homework Solutions
FALL, 2012
Section 5.4
2. Suppose that n is not divisible by any prime number less than or equal to n. Suppose
t
n is composite. Then = pa1 pa Since p1 > n, p2 > n. A contradiction. Thus
n
t.
1
1
a1 =1. But then p1 > n
Math 368
Solutions for Homework due 27 November
Fall, 2012
Note: [x] is the same as x.
1. Computations using Fermats Little Theorem or Eulers generalization
of it.
(a) (54) = (2 33 ) = (2 1)(32 )(3 1) = 18.
(b) (84) = (23 3 7) = 22 (2 1)(3 1)(7 1) = 48.
(
Paul J. Bruillard
MATH 220.970 Problem Set 6
An Introduction to Abstract Mathematics R. Bond and W. Keane
Section 3.1: 3b,c,e,i, 4bd, 6, 9, 15, 16, 18c,e, 19a, 20, 21b
Section 3.2: 1f,i, 2e, 6, 12e,f,h, 13e, 17, 20, 21, 22, 26, 29, 30
Section 3.1
Problem
Derek Damian
Section 903
Proposition 2.1 a
Let n +
For all aZ, aa mod n.
Proof
Let aa mod n, and a, so by reflexivity we see that every element in a is related to itself, aRa.
This is known as reflexivity.
QED.
Derek Damian
Section 903
Proposition 2.1 b
L
Derek Damian
Section 903
Let n +
For all aZ, aa mod n.
Let aa mod n, and a, so by reflexivity we see that every element in a is related to itself, aRa. This is
known as reflexivity.
Derek Damian
Section 903
Let n +
For all a, b Z, if ab mod n, then ba mod
Derek Damian
Section 903
Proposition 3.1
Let a, b, c, d and n be integers with n >0. If ab mod n and cd mod n, then acbd mod n.
Proof:
If ab (mod n) and cd (mod n), then n is contained in (a-b) and n is contained in (c-d), so a - b =
nr and c - d = ns for
Derek Damian
Proposition 4.1
Express 12! as a product of primes.
12! is defined as 12*11*10*9*8*7*6*5*4*3*2*1, so to express 12! as a product of primes, we
write
12!=11*7*52*35*210*1
QED.
Derek Damian
Proposition 4.2
Prove: Let p be a prime and let a and
Math 220
Exam 1 Partial Solutions
February 15, 2012
S. Witherspoon
1. Consider the statement: For all integers m and n, if m is odd, then mn is odd.
(a) [5 points] Write the converse of this statement.
For all integers m and n, if mn is odd, then m is odd
THEOREM OF THE DAY
Mathematical Symbols
Below are brief explanations of some commonly occurring symbols in mathematics presented in more or less haphazard order (the list is not intended to grow so long as to make this irksome). A word of caution -
Mathematical Symbols
Good Problems: April 16, 2004
You will encounter many mathematical symbols during your math courses. The table below provides you with a list of the more common symbols, how to read them, and notes on their meaning and usage. T
MAT067
University of California, Davis
Winter 2007
Some Common Mathematical Symbols and Abbreviations (with History)
Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (January 21, 2007)
Binary Relations
= (the equals sign) means "is the same as"
Useful Mathematical Symbols and Expressions
Symbol AB f () : A B | [a, b] [a, b) P (A) / 0
Name belongs to for all there exists subset product of sets function mapping given that closed interval right-open power set empty set infinity and or
Math 220 October 9, 2003 To PROVE a statement of the form. In each case below, we assume we want to prove a statement of the given form. Some forms can be handled by more than one technique. P Q: Prove both P and Q. P : Usually this comes in the f
Making Mathematical Notes
1
Notes On Making Mathematical Notes For Your Course
C.T.J. Dodson, School of Mathematics, Manchester University
This is a summary of guidance notes for students on mathematical language, symbols, logic and proofs, and on
A List of Tautologies
1. 2. 3. 4. 5. 6. P P (P P ) P P a) P (P P ) b) P (P P ) P P a) (P Q) (Q P ) b) (P Q) (Q P ) c) (P Q) (Q P ) a) (P (Q R) (P Q) R) b) (P (Q R) (P Q) R) a) (P (Q R) (P Q) (P R) b) (P (Q R) ((P
Fall 2003 Math 308/501502 1 Introduction 1.D Autonomous Equations, Stability, and the Phase Line c 2003, Art Belmonte Fri, 05/Sep
Summary
An equilibrium point to which some nearby solutions are attracted and from which others are repelled is called
August 29, 2007 Math 366-Logic Handout In mathematics, it is very important to be able to show that something is correct. This occurs even in elementary school when one shows that six five's is the same number as five six's, or that one piece of a pi