Math 8 Spring Quarter 2015
Homework 2
Due: Tuesday 14 April
1. Let A be the set cfw_a, cfw_1, a, cfw_4, cfw_1, 4, 4. Which of the following statements are true and which
are false?
(a) a A.
(b)cfw_a A.
(c) cfw_1, a A.
(d) cfw_4, cfw_4 A
(e) cfw_1, 4 A
(f)
Math 8 Winter Quarter 2016
Homework 2
Due: Wednesday 20 January
1. Let A be the set cfw_a, cfw_1, a, cfw_4, cfw_1, 4, 4. Which of the following statements are true and which
are false?
(a) a A.
(b)cfw_a A.
(c) cfw_1, a A.
(d) cfw_4, cfw_4 A
(e) cfw_1, 4 A
Math 8 Homework #1 Solutions
1. Consider a mathematical theory in which a statement is any string of
letters where a letter is either a lowercase letter or a capital S. The only
axiom is: S. The logic works like this: in any theorem, any occurrence
of S c
Homework 1 Solutions
Kyle Chapman
October 26, 2012
9.1
True, True, False, True, False, False, True, False, True
9.2
a
D and E
b
You can only deduce that it is not D.
9.3
a
Not valid, would be D C then D C .
b
Valid, this is contraposition. C D then D C
c
Math 8 Fall Quarter 2013
Homework 1
Solutions
1. (i) Prove that the product of two consecutive integers is an even number.
Solution: Let a, a + 1 denote a pair of consecutive integers. Suppose a is even. Then there exists
k Z such that a = 2k. Thus a(a +
Math 8 Homework #2
Due: Friday, April 15th in class
Instructions: I strongly recommend that you attempt all problems on
your own before consulting a classmate or myself. All assignments must be
written individually; no duplicates! Finally, neatness/presen
Math 8 Fall Quarter 2013
Homework 2
Solutions
Number Systems
1. Find geometrically on the line of real numbers the points that represent the following numbers:
(i) 2/5, (ii) 10.
Solution: For nding the precise positions of rational numbers on the real li
Homework 2 Solutions
Kyle Chapman
October 15, 2011
18.1
a
Seeking a contradiction, suppose 3 is rational. This means it can be written as p/q for some integers p and
q . Without loss of generality, assume p and q have no common factors, since we can reduc
Math 8 Homework #2 Solutions
1. Give a direct proof of the following.
(1)
(2)
(3)
(4)
SP
P (G R)
G
P T
S T
Solution.
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(G R) P,
(G R) P,
G R,
P
S
T
S T
Contrapositive of (2).
De Morgans Law on (5).
By (3).
Modus Ponens with (6)
Math 8 - Solutions to Home Work 2
Due: October 11, 2007
1. Every/Only. Sometimes sentences with the words only and every can be conditional statements in disguise. For example, Every even number is a multiple of two.
can be rephrased as If a number is eve
Math 8 - Midterm 1 Solutions
October 19, 2007
1. (12 pts) Consider the proposition R
If I go surng or take a nap, then I will not go surng or I will not take a nap.
(a) (2 pts) Express this proposition symbolically in terms of propositional variables
P an
Math 8 Spring Quarter 2015
Practice Problems for Midterm
No notes or calculators are permitted on this exam. To obtain full midterm credit you must show
all work, provide complete proofs, and provide short by complete explanations for all questions in
the
Math 8
Homework 7
Due in Lecture on Wednesday, August 10th
1. Show that the following are groups:
(a) M22 (Z) = cfw_2 2 matrices with entries in Z under addition.
(b) GL2 (Q) = cfw_2 2 matrices with entries in Q under matrix multiplication.
2. Write out t
Math 8
Homework 1
Due in Lecture on Monday, June 27th
1. Write out the truth table for (P Q) P ) P .
2. (a) Let R, S, T , and W be the following statements:
R = Ara is at the beach
S = Ara is sleeping
T = Ara is doing math
W = Ara is swimming
Using connec
Math 8
Homework 5
Due in Lecture on Wednesday, July 27th
1. Prove that for all m, n, l Z, gcd(m, gcd(n, l) = gcd(m, n, l). Hint: (a b) (b a) a = b
2. Prove the triangle inequality for n elements, which states that for all x1 , x2 , . . . , xn R,
n
n
X
X
MATH 118B MIDTERM PRACTICE SOLUTIONS
1. Short answer questions A.
[5] (a) Give a truth table for (P Q) R
P
T
T
T
T
F
F
F
F
Q
T
T
F
F
T
T
F
F
R P Q (P Q) R
T
T
T
F
T
F
T
F
T
F
F
T
T
T
T
F
T
F
T
T
T
F
T
F
[5] (b) Write S T S \ T as a proposition involving o
Math 8
Extra Credit Problems
Due in Lecture on Monday, August 1st
Remember: Each problem is worth 5 points. You may only use the book or talk to Naomi or me. You
may use any results that we proved in class or that was assigned in homework but you must exp
Math 8
Homework 4
Due in Lecture on Wednesday, July 20th
1. In American football, you can score by getting a safety (2 points), a field goal (3 points), a touchdown
(6 points), or a Point(s) After Touchdown (1 or 2 points, but PAT must follow a touchdown)
Math 8: Transition to higher mathematics
Syllabus
Text: How to prove it (A structured approach) by D. J. Velleman, Second
Edition
Lectures: TR 12:30-1:45 ARTS 1353
Discussions : MW 5:00-5:50 HSSB 1223
MW 6:00-6:50 HSSB 1207
Instructor: Eleni Panagiotou
Of
Math 8
Extra Credit Solutions
Remember: Each problem is worth 5 points. You may only use the book or talk to Naomi or me. You
may use any results that we proved in class or that was assigned in homework but you must explicitly write
when you are doing so.
Math 8
Homework 3
Due in Lecture on Monday, July 11th
1. Courtesy of the movie Die Hard 3: Say only you have a 5-gallon jug and a 3-gallon jug and an
unlimited supply of water. If youre a careful pourer, is it possible to fill the 5-gallon jug with exactl
Math 8
Homework 2
Due in Section on Tuesday, July 5th
1. Given n N, let be the relation on Z defined by a b n|(b a) (recall that n|m means n
divides m, which means d Z such that nd = m). Prove that is an equivalence relation on Z. What are
the equivalence
PSTAT 172 A
Homework Week 1: Intro. to insurance
Ian Duncan, FSA FIA FCIA FCA MAAA
Duncan@pstat.ucsb.edu
University of California, Santa Barbara Note: this weeks homework is unusual in that there are no computations; it is all
written question/answer. Imp
University of California, Santa Barbara,
Department of Mathematics
Math8:IntroductiontoHigherMathematics
Summer2016
Instructor:
OfficeHours:
Website:
Lecture:
GaroSarajian
GSarajian@math.ucsb.edu
W3:005:00pmorbyappointment(emailme)inSouthHall6432F(Pink
si
SYLLABUS FOR MATH 8:
TRANSITION TO HIGHER MATHEMATICS
SPRING QUARTER, 2015
Course information
Instructor: Dr. Lee Kennard
Email: kennard@math.ucsb.edu
Web: www.math.ucsb.edu/kennard (all materials will be posted to the GauchoSpace page)
Meeting times: MWF
PSTAT 172 A
Life Tables
Ian Duncan, FSA FIA FCIA FCA MAAA
Duncan@pstat.ucsb.edu
University of California, Santa Barbara
1
Life Tables (DHW Ch. 3)
A life table is a table of numerical values of the survival function (S0(x) for
discrete integral values of x
PSTAT 172 A
Week 1-2 Survival Models
January 9th 2017
Ian Duncan, FSA FIA FCIA FCA MAAA
Duncan@pstat.ucsb.edu
University of California, Santa Barbara
1
Survival Models (DHW Ch. 2)
In this part of the course we look at Survival Models. A survival model is
PSTAT 120C:
Information and Class Policies
Winter 2017
Instructor
Drew Carter Office hours: Wednesday 1:302:30, Friday 11:0012:00.
5507 South Hall.
email: carter@pstat.ucsb.edu
TAs
Osvaldo Israel and Brian Wainwright
Office Hours: to be announced
Web Page
LECTURE NOTES FOR MATH 8, WEEK 3
S. SETO
1. Proof Techniques
1.1. P Q. To provide a mathematical proof, one begins with a certain set of assumptions, called
hypothesis and with a chain of logical deductions, arrives at a conclusion. Suppose we want to pro
LECTURE NOTES FOR MATH 8, WEEK 4
S. SETO
1. Relations
Definition 1.1. Suppose A and B are sets. Then the Cartesian product of A and B, denoted A B,
is the set of all ordered pairs in which the first coordinate is an element of A and the second element is
LECTURE NOTES FOR MATH 8, WEEK 2
S. SETO
1. Review from last time
1.1. Two methods to show set theory identity. Two sets X, Y are equal if and only if X Y and
Y X, i.e., they are subsets of each other.
Example 1.1. Say we want to show (A B) A = A. We can
LECTURE NOTES FOR MATH 8
S. SETO
1. Introduction
What is mathematics? Up until this point, mathematics courses has been computationally heavy.
In principle, one could have memorized the correct algorithms or formulas, applied and compute some
desired quan
LECTURE NOTES FOR MATH 8, WEEK 4
S. SETO
1. Equivalence relation
Recall that an equivalence relation is a (binary) relation on a set A that is reflexive, symmetric, and
transitive.
Example 1.1. Which of the following relations on R are equivalence relatio