Yong Zhao, Yu-Tzeng Chen, Gouri Gupta
MAT 115A
SSI16
24th, June, 2016
Discussion 1
5 (c) Let a, n, m N with a, n, m > 1. Find a factorization for anm 1.
Proof :
we claim: anm 1 = ( an 1)( an(m1) + an(m2) + an(m3) + . + an + 1)
( a n ) m 1
proof: first con
17 October 2016
CW. Find the first 5 terms of the given seq.
a. An = 2n+1 = 3,5,9,17,33
b. Bk=7+3k = 10,13,16,19,22
c. An=3a2n-1-an-2 = a0=1, a1=2
Ex. Find first 5 terms
An=-5an-1 a0=2
a1 = -5(2) = -10 , a2 = 50 , -250, 1250
DEF: A sequence is called a so
05 OCT 16
2.2 (cont.)
AUB=cfw_x: x A U x B
A
B= cfw_x: x A
x B
= cfw_x: x A
A
A-B =cfw_x: x A
x
B
Ex. Show that
A B A B
Want to show 1.
A B A B
and 2.
A B A B
PROOF 1.
Let
If
x A B
x A
then
then
x A
x A B
. So
so
x A B
( x A x B ) ( x A) ( x B)
. If
x
Yong Zhao
MAT 115A
SSI16
July, 11th
Homework 3
1. (3.7.2) For each of the following linear Diophantine equations, either find all solutions
or show that there are no integral solutions.
1. 60x + 18y = 97
2. 21x + 14y = 147
Proof :
1. 60x + 18y = 97 has no
Yong Zhao
MAT 115A
SSI16
July, 11th
Homework 3
1. (3.7.2) For each of the following linear Diophantine equations, either find all solutions
or show that there are no integral solutions.
1. 60x + 18y = 97
2. 21x + 14y = 147
Proof :
1. 60x + 18y = 97 has no
Yong Zhao
MAT 115A
SSI16
Tuesday, June 28
Homework 1
1. (1.1.5) Prove that
3 is irrational using the Well-Ordering Principle.
Proof :
For sake
of
contradiction,
suppose
3
is
rational,
so
3Q
Then 3 =p/q for some
p, q Z
and q 6= 0.
Note p = q 3. Let S = c
Yong Zhao, Yu-Tzeng Chen, Gouri Gupta
MAT 115A
SSI16
24th, June, 2016
Discussion 2
3. (a) prove there are no integral solutions to equation 13x + 39y = 3 (b)find all integral
solutions to the linear Diophantine 45x + 75y = 60
Proof :
(a)
13x + 39y = 3 has
Yong Zhao
MAT 115A
SSI16
July, 24th
final 2
1. The message in buddha.txt was encrypted using an affine cipher. Decrypt the message.
(1). (20 points) First, provide your key, and show the work explaining how you got your
key.
(2). (10 points) Convert the c
Yong Zhao
MAT 115A
SSI16
July, 11th
Homework 5
1. (6.3.1abc) 1.
For all n cfw_6, 9, 10, calculate (n).
2.
For all n cfw_6, 9, 10, find a reduced residue system modulo n.
Proof :
1.
by the Eulor pld function theorem,
when n = 6, (6) =| cfw_1, 5 |= 2
when n
Patrick Weed
MAT 201A
Babson
25 October 2013
Homework 4
2. (Homework 3) Let p : R2 R be a polynomial function of two real variables. Suppose that p( x, y) 0 for all x, y R. Does every such function attain its infimum? Prove
or disprove.
Claim: Not every s
Website Review example
Summary of findings
This report discusses a review of the User Experience (UX) of the Your Site Here
website. A number of areas were found where improvements can to be made to the
site design, navigation, and content.
The navigation
Yong Zhao
MAT 115A
SSI16
July, 11th
Homework 4
1. (4.3.2) Find an integer that leaves a remainder of 1 when divided by either 2 or 5, but
that is divisible by 3.
Proof :
since we need an integer that leaves a remainder of 1 when divided by either 2 or 5,
Math 261
Insert Name Here
Fall 2010
Prof. Belk
Sample Assignment 2
Exercise 1.5.11
Part (1)
(1) (x in U )[R(x) C(x)]
(2) (x in U )[T (x) R(x)]
Consider an arbitrary a in U .
(3) R(a) C(a)
(1), Universal Instantiation
(4) T (a) R(a)
(2), Universal Instanti
Yong Zhao
MAT 115A
SSI16
July, 11th
Homework 5
1. (6.3.1abc) 1.
For all n cfw_6, 9, 10, calculate (n).
2.
For all n cfw_6, 9, 10, find a reduced residue system modulo n.
Proof :
1.
by the Eulor pld function theorem,
when n = 6, (6) =| cfw_1, 5 |= 2
when n
Yong Zhao
MAT 115A
SSI16
July, 11th
Homework 3
1. (3.7.2) For each of the following linear Diophantine equations, either find all solutions
or show that there are no integral solutions.
1. 60x + 18y = 97
2. 21x + 14y = 147
Proof :
1. 60x + 18y = 97 has no
HOMEWORK 2, MATH 150A, FALL, 2016
Due date: Monday, 10/3/2016, 10AM
Suggested readings: Artin, 2nd Edition 2.1-2.4
(1) Let X be the set of real numbers excluding 1. Define ? on X by x ? y = x + y + xy.
Check that (X, ?) is a group.
(2) Let R be the set of
HOMEWORK 3, DUE 10/10/2016, MONDAY 10AM, MATH 150A
Suggested Readings: Sections 1.5, 2.1-2.6
(1) (a) As r varies, show that (123 n)r (12)(123 n)nr gives all the transpositions
(12), (23), (34), ., (n-1 n), (n 1). (b) Show that any transposition in Sn is a
03 OCT 16
Show for any prime number P if P divides A, A
Z, then P does not divide A+1
PROOF: Let A Z and P be a prime number s.t. P divides A. Then A=PK for some K Z. Suppose by
contradiction that P divides A+1, then A+1 =P*M where M Z.
Then 1=(A+1)-1=PM