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
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
Math 101: Midterm I
October 4th, 2013
1. Circle T or F, depending on the truth value of the following statements. For statements with
quantifies the universal set is R.
a)
A 6= cfw_A.
b)
(x)(y)(x y = 0).
c)
(y)(x)(x y = 0).
T
d)
(x)(y)(xy = 0) (x + y = y)
Practice Midterm 2
1. Circle T or F, depending on the truth value of the following statements. For statements with
quantifies the universal set is R.
a)
The relation on R given by xRy if xy > 0 is an equivalence relation.
b)
If the set A is finite and adm
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
Selected Solutions to Homework from Section 2.2, 2.3, and 2.4
2.2.9.a. Let A and B be sets. Prove that A B i A B = .
Proof. Let A and B be sets.
: By contrapositive. Suppose A B is not empty. Then there exists x A B. Then x A
and x B. Then it is not the c
Practice Midterm Exam
Math 108 Fall 2014
Please write your name, student ID number, and test version on the front of your blue book.
Please answer the questions in the blue book, clearly labeling the problem number. Write neatly
and clearly. If you have s
Midterm Exam 1B
Math 108 Fall 2014
Please write your name, student ID number, and test version on the front of your blue book.
Please answer the questions in the blue book, clearly labeling the problem number. Write neatly
and clearly. If you have some sc
Midterm Exam 1A
Math 108 Fall 2014
Please write your name, student ID number, and test version on the front of your blue book.
Please answer the questions in the blue book, clearly labeling the problem number. Write neatly
and clearly. If you have some sc
Math 101: Midterm I
October 4th, 2013
1. Circle T or F, depending on the truth value of the following statements. For statements with
quantifies the universal set is R.
a)
A 6= cfw_A.
T F
b)
(x)(y)(x y = 0).
T F
c)
(y)(x)(x y = 0).
T F
d)
(x)(y)(xy = 0) (
Math 108, Fall 2013.
Oct. 25, 2013.
MIDTERM EXAM 1
. . KEY
NAME(pr1nt 111 CAPITAL letters, rst name rst): _
NAME(sign): _
ID#: _
Instructions: Each of the 5 problems has equal worth. Read each question carefully and answer
it in the space provided. YOU MU
Fu Liu
Math 108
SOLUTIONS TO HOMEWORK 6
3.2.1
(c) Reflexive on N, symmetric, and transitive.
(d) Transitive.
(f) Symmetric.
(l) Symmetric.
3.2.2
(b) cfw_(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)
(c) cfw_(1, 2), (2, 1).
3.2.4
(c) Proof. i. Assume R is transit
Fu Liu
Math 108
SOLUTIONS TO HOMEWORK 2
1.3.1
(b) (x)(x is precious x is not beautiful.)
(c) (x)(x is isosceles x is a right triangle.)
(e) (x)(x is not isosceles x is a right triangle.)
(f) (x)(x is honest) (x)(x is honest.)
1.3.2
(b) (x)(x is precious x
HOMEWORK ASSIGNMENT 5 - MAT 108
DUE: FRIDAY, NOVEMBER 18 AT 3:10 PM
(1) Let f : A B and g : B A. Prove that if g f = IA and f g = IB , then f and g are bijections.
(2) Prove that if f : A B and g : C D are bijections, then the function f g is also a bijec
HOMEWORK ASSIGNMENT 4 - MAT 108
DUE: WEDNESDAY, NOVEMBER 9 AT 3:10 PM
(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 equival
HOMEWORK ASSIGNMENT 3 - MAT 108
DUE: WEDNESDAY, OCTOBER 26 AT 3:10 PM
(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.
Base Case:
n = 1 : (1
HOMEWORK ASSIGNMENT 2 - MAT 108
NAME: TRISTAN BALA (912265535)
(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.
The contrapositive of this proposition is: if x divides z, then x
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)
(2) Which of the following pairs of propos
HOMEWORK ASSIGNMENT 2 - MAT 108
(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. (by contraposition) Let x, y and z be any integers. Now, consider the statement if x|z,
t
Math 108, Fall 2013.
Oct. 25, 2013.
MIDTERM EXAM 1
NAME(print in CAPITAL letters, rst name rst):
NAME(sign):
Instructions: Each of the 5 problems has equal worth. Read each question carefully and answer
it in the space provided. YOU MUST SHOW ALL YOUR WOR
HOMEWORK ASSIGNMENT 4 - MAT 108
(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
partition of A.
(
HOMEWORK ASSIGNMENT 5 - MAT 108
(1) Let f : A B and g : B A. Prove that if g f = IA and f g = IB , then f and g are bijections.
Proof. Assume f : A B and g : B A and assume g f = IA and f g = IB . First we will show
IA and IB are bijections. Note that sin
HOMEWORK ASSIGNMENT 3 - MAT 108
(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.
Proof. Base Case: If n = 1 then 1 1! = 1 and (n + 1)! 1 = (
Math 108, Fa112013.
Nov. 22, 2013.
MIDTERM EXAM i
NAME(print in CAPITAL letters, rst name rst):
NAME(sign):
Instructions: Each of the 5 problems has equal worth. Read each question carefully and answer it
in the space provided. YOU MUST SHOW ALL YOUR WORK
Justification for Proof by Contradiction
Proof by contradiction for the proposition P follows these basic steps:
1. Assume P is false. That is, assume P is true.
2. Show that this leads to a contradiction.
Justification for this can be shown in the follow
HOMEWORK ASSIGNMENT 1 - MAT 108
(1) Make a truth table for each of the following propositional forms:
(a) (P Q)
P
T
T
F
F
Q P Q (P Q)
T
T
F
F
F
T
T
F
T
F
F
T
(b) (P Q) (P R)
P
T
T
T
T
F
F
F
F
Q R
T T
T F
F T
F F
T T
T F
F T
F F
P Q P R
T
T
T
F
F
T
F
F
F
F
L4 Quantified Statements
Math 108, W 2017
Last time 1.1-1.2
This time 1.3
Next time: 1.4-1.6 (proof methods)
Monday 2.1-2.3 (Sets)
Wednesday 2.4-2.5 (induction)
a week from that is MT1
Pre-recs
Today we study predicate logic, another model of some aspect
L3 Computation in Truth Tables;
Math 108, W 2017
Last time 1.1-1.2
This time 1.1-1.2
Next time 1.3
Caveats
First:
This is not the main idea of the class; the language of (definitions, theorems, and) proofs is the main
idea of the class.
Truth tables help