1
Math 3040
Prelim 1: Solutions
Lecture 1
February 24, 2011
Problem 1: Give an example of logical expressions A and B, each of which is expressed
in terms of atomic statements P and Q, such that
a. A B is a tautology.
b. B A is not a tautology.
Justify yo
Math 3040: Solutions for Homework 8. (1) For each of the following relations on the set of integers Z = cfw_0, 1, 1, 2, 2, 3. determine: is it reexive? Is it symmetric? Is it transitive? is it an equivalence relation? Prove your claims. (a) (a, b) R i 1 a
MATH 3040
Assignment 5
1. Let A be the left set; let B be the right set. We have to show that A B and B A.
To show that A B, let x A. Then x a and min(x, a) b. Either a b or
b a. If a b then x a = min(a, b), thus x B. For b < a, we know that x b
since x a
MATH 3040
Assignment due April 24, 2014
Projective geometry and the extended Euclidean plane
From Hilberts treatment of the axioms of Euclidean plane, a completely worked out axiom
system for geometry in the plane is quite complicated. By the nineteenth c
Math 3040, Prelim I
February 27, 2014
Please use two exam booklets, and turn them both in at the end of the Prelim. One booklet
is for you nal write-up of your answers and proofs. The other is for scratch work, where
you can do any calculations and organi
MATH 3040
Assignment for Week 10
1. Consider the set S = cfw_na + mb | n, m 0, n, m Z, for xed a, b N.
Prove that if a, b are relatively prime, then ab a b S. Note that a, b are positive
/
integers. Hint: Show that if ab a b = na + mb, then n 1 (mod b) an
MATH 3040
Assignment 6
1. 1 < 2 by Theorem 4. Still by Theorem 4, #N < #P(N). And since #R = #R(N),
we have 0 < 1 . Therefore 0 < 1 < 2 .
2. X0 = cfw_2n + 1 | n N
3. C = (, 1]
4. The cardinality of cfw_(n1 , n2 , . . . , nk , . . .) | nk N is 1 . Try to s
Math 3040, Some Probability Problems
Due March 20, 2014
The following problem is from Pillow Problems and a Tangled Tale by
Lewis Carroll.
Problem: A bag contains 2 counters, as to which nothing is known except
that each is either black or white. Ascertai
Math 3040, Take Home Prelim 2
Due April 29, 2014
Please hand in your solutions as a TeX document in class on Tuesday April 29, 2014, and be sure
to cite the sources that you use in your answers.
1. In your own words, state the precise denition of what the
MATH 3040
Assignment 9
1.
(a) Let a, b, c 2 N. We dene that g 2 N is the greatest common divisor of a, b, and c
and denote it by gcd(a, b, c), if both the following hold: i) g divides a, b, and c, ii)
whenever d 2 N divides a, b, and c then d also divides
MATH 3040
Assignment 7
1. Let P (a), P (g), and P (p) be the probability of the student choosing art, geology, and
psychology, respectively. Then we have P (a) + P (g) + P (p) = 1 since the student
has to choose one of them, and P (a) = P (p) = P (g)/2. H
Math 3040, Some Set Theory
Due March 13, 2014
Recall that a function f : X Y is injective if for all x1 , x2 X, f (x1 ) =
f (x2 ) implies x1 = x2 , and f is surjective if for all y Y , there is an x X
such that f (x) = y. A function g : Y X is a right inv
MATH 3040
Assignment 8
1. Let d be the diameter of a quarter. Then the edge of each square of the checkerboard
is 2d. In order to win this game, the center of the quarter must be at least its radius
d/2 away from all the edge of the square of the checkerb
MATH 3040
Assignment 4
1. Let m = 2k + 1, n = 2q + 1 for some k, q Z. Then n2 m2 = (2q + 1)2 (2k + 1)2 =
4(q k)(q + k + 1). Since q k and q + k are either odd or even at the same time,
we know 2|(q k)(q + k + 1). Therefore n2 m2 is divisible by 8.
2. Supp
Math 3040:
Spring 2011
Impossibility Proofs
Contents
1. Rings, Fields, and Vector Spaces
2. Polynomials
3. Fields and Polynomials
4. The Impossibility of Certain Ruler and Compass Constructions
4.1. Geometric constructions
4.2. Constructing numerical quan
Math 3040:
Spring 2011
The Rational Numbers
Contents
1. The Set Q
2. Addition and multiplication of rational numbers
2.1. Denitions and properties.
2.2. Comments
2.3. Connections with Z.
2.4. Better notation.
2.5. Solving the equations Ea,b and Ma,b .
3.
Math 3040:
Spring 2011
The Integers
Contents
1.
2.
3.
4.
The Basic Construction
Adding integers
Ordering integers
Multiplying integers
1
4
11
12
Before we begin the mathematics of this section, it is worth recalling the mind-set
that informs our approach.
1
Math 3040
Prelim 1: Solutions
Lecture 2
February 24, 2011
Problem 1: Suppose X is a logical expression in terms of atomic statements P and Q,
and you know only that (P Q) X is a tautology. Using reduced truth tables for P Q,
X, and (P Q) X, show that, u
Math 3040
Prelim 2: Solutions
Lecture 2
Problem 1: This problem will test your ability to do mathematical induction. State which form
of induction you are using, and give a (brief) reason for each step in your induction proof.
Suppose a0 , a1 , a2 , . . .
Math 3040
Prelim 2: Solutions
Lecture 1
Problem 1: This problem will test your ability to do mathematical induction. State which form
of induction you are using, and give a (brief) reason for each step in your induction proof.
The factorial function n! is
Math 3040, Quick Quiz: Solutions
March 15, 2012
1. Lets play a game. Start with the number 1000. You then subtract any
integer from 1 to 100 inclusive and announce that number. Then I take
your number, select a integer from 1 to 100 inclusive and subtract
Math 3040 Prove it!
Spring 2012
Solution to selected problems from Homework 7
4.3.13 Let A and B be sets and let f, g : A B be functions. Prove that
there exists a set E and a function e : E A such that f e = g e, and
that for any set C and function h : C
Math 3040 Prove it!
Spring 2012
Solution to selected problems from Homework 8
5.1.8 Let A be a set, and let R be a reaction on A.
(1) Suppose that R is reexive. Prove that
xA [x]
= A.
(2) Suppose that R is symmetric. Prove that x [y] if and only if y [x],
Math 3040 Prove it!
Spring 2012
Solution to selected problems from Homework 10
Extra 1 Find a closed form expression for the sum of the fourth powers of
the integers from 1 to n, where n is a positive integer.
Proof. We know (k + 1)5 k 5 = 5k 4 + 10k 3 +
MATH 3040:
PROVE IT!
Peter Kahn
Spring 2011
Introduction
1. The role of proof
Mathematics pervades human culture. From the simplest daily transactions to the
largely hidden mathematical concepts that underlie the design and operation of much
of our infras
Math 3040
Spring 2011
Glossary of Logical Terms
The following glossary briey describes some of the major technical logical
terms used in this course. The glossary should be read through at the beginning and can then be consulted again as needed. The organ
Math 3040
Spring 2011
The Predicate Calculus
Contents
1. Introduction
2. Some examples
3. General elements of sets.
4. Variables and constants
5. Expressions
6. Predicates
7. Logical operations and predicates
8. Specialization
9. Logical equivalence of pr
Math 3040
Spring 2011
The Natural Numbers
Contents
1.
2.
3.
4.
5.
6.
7.
8.
History
The basic construction
Proof by Induction
Denition by induction
Addition
Addition and the natural ordering
Variants on induction
Multiplication
1
2
5
8
9
12
13
15
Proofs ar