Section 1.3 : Edge Counting
Example: Prove that in any graph, the sum of the degrees of all vertices is
equal to twice the number of edges.
Example: Suppose we want to construct a graph of 20 edges
Section 2.2 : Hamilton Circuits and Paths
Denition: A Hamilton circuit is a circuit that Visits each vertex in a graph
exactly once (except the start / end vertex).
Denition: A Hamilton path is a p
Section 2.3 : Graph Coloring
Denition: A coloring of a graph G assigns colors to the vertices of G so that
adjacent vertices are given different colors.
Denition: The minimal number of colors requi
Section 2.4 : Coloring Theorems
Theorem 1: A graph G is 2colorable if and only if all circuits have even length.
Proof: Recall that a graph G is bipartite if and only if every circuit in G has
even l
Math 3034
1
Logic
Truth tables:
and (conjunction)
or (disjunction)
P
T
T
F
F
P Q
T
T
T
F
Q
T
F
T
F
P
T
T
F
F
Q
T
F
T
F
P
T
T
F
F
P
T
F
P
F
T
if and only if [iff] (equivalence)
P Q
T
F
T
T
Q
T
F
T
F
Review 3 : Sections 1.1-1.4, 2.1-2.4 : Summary
Chapter 1
Results
1. If I is an independent set, than V
2. If C is an edge cover, than V
I is an edge cover.
C is an independent set.
3. G1 and G2 are is
Review 2 : Sections 1.1-1.4, 2.1-2.4 : Part 2
Example: Is the graph in exercise 4d on page 64 planar? Prove it.
Example: Is the graph in exercise 3L on page 39 planar? Prove it.
Example: Does the grap
Homework #12
1. Prove by induction on n: If n 2, then
1
1
4
1
1
9
1
1
n2
=
n+1
.
2n
Note: On the left-hand side is a product, not a sum. First do the base case. Then write down the k -case and say w
Homework #14
1. For every real number r, set Ar = cfw_(x, y ) R2 : y = x2 + r.
(a) Give an elementwise proof that R2
r R
Ar .
(b) Give a contrapositive proof of the following statement:
Statement: Fo
Practice for Test #2
MATH 3034, Fall 2012
1. Let a, b, m1 , n1 , m2 , and n2 be integers such that
m1 a + n1 b = 60;
m2 a + n2 b = 45; and
3 divides a and b, but 3 = gcd(a, b).
Determine (with justica
Sample quiz
MATH 3034, Spring 2012
Show all your work. No calculators allowed.
1. Show that the following two statements are logically equivalent:
P (Q R)
and
(P Q) R
Do this without lling in a truth
Section 1.1 : Graph Models
Denition: A graph G = (V, E) consists of a nite set V of vertices and a set
E of edges joining dierent pairs of distinct vertices.
Denition: A path P is a sequence of distin
Review 2 : Sections 1.1-1.4, 2.1-2.4 : Part 1
Example: Find a maximal independent set and a minimum edge cover for the
graph in Figure 1.2 on page 5. Prove that no larger independent set exists.
Examp
Math 3034
1
Proofs
Day 1-1
Day 1
Syllabus
Canvas
LaTeX
What is Math?
Clarity
Self-Explanation
Example proofs
Nicholas Robbins. Compiled: August 24, 2017
Page 1 of 6
Day 1-1
Proofs
Question:
Mat
Math 3034
1
Quantifiers
Day 2-2
Quantifiers
Definition 2.3.8 (Universal Quantifier):
Let
P (x) be an open statement with universal set Ux , then the statement:
is defined to be true exactly if the
tru
Math 3034
1
Equivalences, Normal Forms, & Quantifiers
Day 1-3
Tuatology
Question:
What are the possible truth values of (P Q) (P Q)?
Definition 2.2.11
A statement form, T , that is
for all truth value
Chapter 3: Direct and Proof by Contrapositive
Problem 3.16. Let x Z. Prove that if 7x + 5 is is odd, then x is even.
Construction. Symbolically, we can express this as
(x Z)[(7x + 5 Zo ) (x Ze )].
The
Chapter 2: Logic
Problem 2.32. In each of the following, two open sentences P (x) and Q(x) over
a domain S are given. Determine all x S for which P (x) Q(x) is a true
statement.
(d) P (x) : x [1, 2];
Math 3034, Introduction to Proofs
These lecture notes are prepared by Tao Lin
based on
Mathematical Proofs
A Transition to Advanced Mathematics
by G. Chartrand, A.D. Polimeni, and P. Zhang
A Major Goa
REVIEW FOR FINAL EXAM MATH 3034
1. In each of (a) i (d) give, in written form, both the negation and the contrapositive
of the given implication. (Label your answers as the negation and the contraposi
1. (a) A set F of functions on R is NOT equicontinuous at a point x0 R provided
there exists a positive real number such that for all positive real numbers
there exists a function f F and a real numb
Test 1 Review - Class
1. The following items give denitions of terms. Tell me (in english) what the
denition is for the OPPOSITE of the dened term. For example, if the term
is transitive .what does it
Review for Test 2 Part 1
1. Given A = cfw_ , 1, 2, cfw_1, cfw_2 , B = cfw_ cfw_1, cfw_2, cfw_1, 2, and C = cfw_
Mark each of the following true or false.
a) A
C
b) B
C
c) B
A
d) A
B=C
e)
A
f)
, 1, 2,