Math 110
Proof and Mathematical Reasoning
Jenny Wilson
This package is optional reading for Math 110 students. It is highly recommended for students with less
prior proof-writing experience.
A Primer on Mathematical Proof
A proof is an argument to convinc
MAT 108 Midterm 2 Review
This midterm will cover Sections 2.4, 2.5, 3.1, 3.2, 3.3, 4.1, 4.2 and 4.3. In order to do well
in the midterm, you need to know everything we did in class and be able to do all the homework
problems. Here is a list of the main ma
Selected Solutions to Homework from Section 1.6 and 2.1
1.6.5.a. Prove the natural number x is prime i x > 1 and there is no possible integer
greater than 1 and less than or equal to x that divides x.
Comments: This is proof is saying that is sucient to c
HW4
Cherneys Math 108, S 2017
Due 5/5, 12pm, at the class box.
Reading
3.1-3.3
Writing With Mathematics part II: Words (If you have not already read it.)
Exercises
Follow the rules on HW0!
Each exercise should be completed by stating a claim and presentin
HW6
Cherneys Math 108, S 2017
Due 5/19, 12pm, at the class box.
Reading
4.1-4.6
Writing With Mathematics part II: Words (If you have not already read it.)
Exercises
Follow the rules on HW0!
Each exercise should be completed by stating a claim and presenti
HW0
Math 108 S 2017
Due 4/7, 12pm at the class box. A video walk to the class box (where homework is to be handed in)
can be found in the Canvas announcements.
This homework is for extra credit, to be applied to your homework score after the curve for the
HW2
Math 108 W 2017
Due 1/27, 5pm at the class box.
Reading
1.5 and 1.6
2.1-2.4
1.7 and 1.8 are full of examples of proofs and advice on constructing proofs, but will be considered
optional reading.
Exercises
Follow the rules on the next page.
1.4
#5a Use
HW5
Cherneys Math 108, S 2017
Due 5/12, 12pm, at the class box.
Reading
3.4-3.5
4.1
Writing With Mathematics part II: Words (If you have not already read it.)
Exercises
Follow the rules on HW0!
Each exercise should be completed by stating a claim and pres
HW7
Cherneys Math 108, S 2017
Due 5/26, 12pm, at the class box.
Reading
6.1
Writing With Mathematics part III: Formatting
Exercises
Follow the rules on HW0!
1.1#12d
1.2#12d
1.3#1a
2.1#9
2.2#9a
2.4#4e
6.1
#1behk You may either prove that the pair do not fo
HW9
Cherneys Math 108, S 2017
Due 6/9, 12pm, at the class box.
Reading
6.4-6.5
Exercises
Follow the rules on HW0!
6.2#10
6.3#4b
6.4#1ab,3ab,10abc
6.5#2,11ab
Be sure to leave plenty of time to study for the final exam too! It will be on Thursday, June 15th
HW3
Cherneys Math 108, S 2017
Due 4/28, 12pm, at the class box.
Reading
2.5
Writing With Mathematics part II: Words
Exercises
Follow the rules on HW0!
1.2#12b to keep you thinking about the basics of logic
1.5#3c,6d
2.2#10d
2.4#4j,5a
2.5#3
Selected Instru
MT2 Vocaulary
Cherneys Math 108, S 2017
Note that you do need to know the vocabulary from MT1 on midterm 2. This is just a list of the
new vocabulary!
Relation
identity relation
domain (of a relation)
range (of a relation)
inverse (of a relation)
co
Math 25, Fall 2014.
HW 1 Solutions (mostly adapted from Abbotts Instructors Manual)
1.2.1. (a) We prove this by contradiction. Assume that there exist integers p and q satisfying
2
p
= 3.
q
We may assume that p and q have no common factor. From the above
MAT 108
Problem 1
Prof. Fu Liu
Practice 3
Fall 2016
Give the definition of each part below.
(a) A surjection.
Answer: A function f : A B is a surjection (or is onto B) if Rng(f ) = B.
onto
We write f : A B.
(b) Equivalence of two sets A and B.
Answer: Two
HOMEWORK ASSIGNMENT 2 - MAT 108
ALI HEYDARI
(1) Prove the following by contraposition:
(a) For all integers x, y, and z, if x does not divide yz then x does not divide z.
Proof. We know that in order to prove something by contraposition,
if we have P Q th
HOMEWORK ASSIGNMENT 1 - MAT 108
DUE: MONDAY, OCTOBER 3 AT 3:10 PM
(1) Make a truth table for each of the following propositional forms:
(a) (P ^ Q)
(b) (P ^ Q) _ (P ^ R)
(c) P ) (Q ^ P )
(d) (P _ Q) ) (P ^ Q)
(e) (P ^ Q) _ (Q ^ R) ) (P _ R)
Soloution:
par
HOMEWORK ASSIGNMENT 4 - MAT 108
ALI HEYDARI
(1) We know that every partition of a set A is the set of equivalence classes of A modulo some equivalence
relation R, and for every equivalence relation R on a set A the set of equivalence classes forms a
parti
HOMEWORK ASSIGNMENT 1 - MAT 108
ALI HEYDARI
(1) Make a truth table for each of the following propositional forms:
(a) (P Q)
(b) (P Q) (P R)
(c) P (Q P )
(d) (P Q) (P Q)
(e) (P Q) (Q R) (P R)
Solotions: part a)
P Q (P Q)
T T
F
T F
T
F T
T
F F
T
part b)
P Q
Dept. of Computer Science, University of California, Davis
ECS30, Summer Session I 2017
Instructor: Rob Gysel
Sample Midterm
July 13th , 2017
Name:
Student ID:
Do not open the exam until instructed to do so.
You will be given exactly 1 hour and 40 minut
MAT 108
Problem 1
Prof. Fu Liu
Practice 3
Give the definition of each part below.
(a) A surjection.
(b) Equivalence of two sets A and B.
(c) A denumerable set (including the cardinality part).
Fall 2016
Problem 2
Indicate whether following statement is tr
MAT 108
Problem 1
Fu Liu
Practice Exam 2
Give the denition of each part below.
(a) A symmetric relation on a set A.
Answer: Suppose R is a relation on A. Then R is symmetric i for all x and y in A,
if xRy, then yRx.
(b) A partition of a nonempty set A.
An
MAT 108
Problem 1
Fu Liu
Practice Exam 2
Give the denition of each part below.
(a) A symmetric relation on a set A.
(b) A partition of a nonempty set A.
(c) The characteristic function of a set A.
Problem 2
Give the equivalence relation R on A = cfw_1, 2,
HW1
Cherneys Math 108, S 2017
Due 4/14, 12pm, at the class box.
A video walk to the class box (where homework is to be handed in) is in Canvas announcements.
Reading
1.2-1.4
The handout Writing With Mathematics Part 1: Sentences
Exercises
1.1#12c
1.2#1de,
MAT 108
HW #7
(1) (6 points) Are the following sets equivalent
(a) cfw_pt R and unit circle S 1 = cfw_(x, y) R2 : x2 + y 2 = 1?
(b) R2 R and unit cylinder C = cfw_(x, y, z) R3 : x2 + y 2 = 1, z R?
(c) Set of solutions of equation xn + an1 xn1 + + a1 x + a
HOMEWORK ASSIGNMENT 1 - MAT 108
MISSAEL COLIN CUEVAS
DUE: MONDAY, OCTOBER 3 AT 3:10 PM
(1) Make a truth table for each of the following propositional forms:
(a) (P Q)
Answer:
P Q P Q (P Q)
T T
T
F
T F
F
T
F T
F
T
F F
F
T
(b) (P Q) (P R)
Answer:
P
T
T
T
T
HOMEWORK ASSIGNMENT 2 - MAT 108
MISSAEL COLIN CUEVAS
(1) Prove the following by contra-position:
(a) For all integers x, y, and z, if x does not divide yz then x does not divide z.
Answer:
Rewrite contra-position: If x divides z, then x divides yz.
z
= k
HOMEWORK ASSIGNMENT 4 - MAT 108
MISSAEL COLIN CUEVAS
(1) We know that every partition of a set A is the set of equivalence classes of A modulo some equivalence
relation R, and for every equivalence relation R on a set A the set of equivalence classes form
HOMEWORK ASSIGNMENT 3 - MATH 108
MISSAEL COLIN CUEVAS
(1) Prove the following using induction, generalized induction, and/or complete induction:
(a) For all natural numbers n,
1 1! + 2 2! + 3 3! + + n n! = (n + 1)! 1.
Answer:
Suppose n = 1
1 1! = 2! 1, im
MAT 108
HW #6
Assume that all functions in the exercises below are single-valued, i.e. each element
under the mapping has only one image.
(1) (4 points) Prove that if f : A B then |Rng(f )| |A|
(2) (4 points) Let A be an infinite set. Prove that A is equi
MAT 108
HW #10
(1) (4 points) Let G be a group and H be a subgroup of G.
(a) If G is Abelian, must H be Abelian?
(b) If H is Abelian, must G be Abelian?
(2) (4 points) Prove that every subgroup of a cyclic group is cyclic.
(3) (4 points) Let 3Z and 6Z be
MAT 108
HW #8
(1) (4 points) Find two sets A and B with |A| = |B| = 1 such that A B is
(a) empty
(b) denumerable
(c) finite and nonempty
(d) uncountable
(2) (4 points) Prove that ifScfw_Bi : i N is a denumerable family of pairwise disjoint
distinct finite
ENGINEERING 106: Engineering Economics, Winter Quarter 2017
Lecture 3 Objectives:
Learn to play with the F, P, i, N using an algebraic polynomial for a single transaction
(i)
(ii)
(iii)
(iv)
find F with known P, i, N
find P with known F, i, N
find i with
Homework 2
Due Feb. 8, 2017
Department of Computer Science, U.C. Davis
Problem 1 (a)
3k T (
X 3k cn
n
)+
+ cn
k
2
2k
(1,k)
The value of k = log2 n (b)
T (n) = T (n 1) + 1 = (T (n 2) + 1) + 1 = (T (n 3) + 1) + 1 + 1).
(n)
Problem 2
Problem 3
Problem 4 Dete