Assignment 11
u5641446
October 20, 2016
Question 1
(a)
We know (1) is a maximal ideal. Since our ring is a field, we know every ideal is generated by a monic polynomial. We
want to define a map : R[x] R[x]/(x2 ). We define the map such that constants get
Assignment 10
u5641446
October 14, 2016
Question 1
From lectures we know if a ring has nontrivial idempotents, R
= eR (1 e)R for e idempotent. So we check if our ring Z8 has any
nontrivial idempotents.
22 = 4
32 = 1
42 = 0
52 = 1
62 = 4
72 = 1
Hence it
COMP2610/COMP6261  Information Theory: Assignment 3
Andrew Markovic u5641446
October 16, 2016
COMP2610/6261 Shared Questions
1. (20 pts) Let X be an ensemble with AX = cfw_x 1 , x 2 , x 3 and probabilities pX = (1/4, 1/3, 5/12).
(a) (2 pt) Compute the c
COMP2610/COMP6261  Information Theory: Assignment 3
Andrew Markovic u5641446
October 16, 2016
COMP2610/6261 Shared Questions
1. (20 pts) Let X be an ensemble with AX = cfw_x 1 , x 2 , x 3 and probabilities pX = (1/4, 1/3, 5/12).
(a) (2 pt) Compute the c
Assignment 9
u5641446
October 6, 2016
(Question 1)
For this section we are trying to find the group that gets sent to 0. And then the smallest set of elements that generates
that group.
(a)
This function f (x, y) f (0, 0) maps every element to its constan
5. Use properties of limits and algebraic methods to find the limit, if it exists. (If the limit
is infinite, enter ' or '', as appropriate. If the limit does not otherwise exist, enter
DNE.)
limx 0
4.6x2 + 6.2x
x
6./1 points
Use properties of limits and
Inverse Derivative Problems
1. Let f be the function defined by f(x) = x3 + x. If g(x) = f 1 (x) and g(2) = 1,
what is the value of g (2)?
2. Let f be the function defined by f(x) =
4+ x 5
1
3 x 2 . If g(x) = f (x), and g(3)=2,
what is the value of g (3)?
Math 138: Precalculus
Practice Test 1
Name:_
1) Given the points (2,6) and (4,6) find
a. the midpoint of the line segment joining the points
b. the distance between the points
c. find the slope of the line between the points
d. find the equation of the
Math 138: Section 1.5
Analyzing Graphs of Functions
Defn: The Q g oi ax R/ngar are the xvalues where f(x)=O. (i.e., the
xintercepts).
Ex: Find the zeros of the following functions:
3) f(x) = 3x 6
0* 3x4:
(023%
(lgxi
b) x) =x25x6
Xt 95 O
(X 4:1 (15):0
X
Math 138: Section 1.4
Functions
5" i
Defn: A W, dwh f from a set A (xvalues) to a set B (yvalues) is a relation for
which every x has only one y.
Ex: Are the following relations functions?
v:
I
Nol m Funcm lflt. ; v (313 MAM M V
X; Z hid l6 "3?" ' m h
Math 138: Section 1.3
Linear Eguations in Two Variables
H II
Given y = mx + b, m is the slope and "b" is the yintercept (where x = O)
The 61092. between two points (x1,y1) and (352,312) is: m = g = 93" : 1
rum AV X2_Xl
Slopes WSHWLSRP" negaWopo erHLd
Math 138: Section 1.1
Rectangular Coordinates
Defn: The RHEth ulav CODY[Lib QR Him or
Cw Lam
represented (Le. plotted).
Pyjhagorean Theorem:
PlLH m Pen/1J6
UWL Conchrlywo {lack Pod/Hs.
OHM
For (L nmt Jmangut
Z
az+b2fC
mm
b
Plan 5: : the 2D plane such
Math 138: Section 1.2
Graphs of Eguations
Defn: E330 aims will TWO Vwi'ab :3; :The relationship between
two variables. g) Lg, {Uri
A: TTr" In is (mlmix 3J1
{ : xl+ 1
Defn: A point is a éolu'hon to an equation if the statement holds true when you plug
t
Math 138: Section 1.6
A Library of Parent Functions
1) Constant Function: f(x) = c 2) Linear Function:f(x) = x
3) Quadratic Function (Parabola): f(x) = x2 4) Cubic Function:f(x) = x3
f lllxizx
5) Reciprocal Function: f(x) 2%
7) Absolute Value Function
Math 138: Section 1.7
Transformations of Functions
*Many functions have graphs that are simple transformations of the parent graph studied in
section 1.6.
Vertical and Horizontal Shifts:
Let c be a positive real number. forHm and HDYI ZOYih/i shifts in t
Math 138: Section 2.3
Poiynomial and Synthetic Division
I We can use poiynomial long division to factor.
Example: Factor 2614+ 5x3 + 6x2 x 2 by using long division to divide byx + 2
5i2 3; +§x°+ézx5xra
353+GXL )(ferngxtXél '7 (W17ohgx fl)
H 313242)?)
Lx
Math 138: Section 2.1
Quadratic Functions
Defn: Let n be a nonnegative integer and let an, an_1, , a2, £11,110 be real numbers with
an at 0. Then the function given by f(x) = anxl + anxn1 + + azxz + alxl + a0 is the
polynomial function ofdegree n.
Note
Math 138: Section 1.8:
Combinations of Functions: The Composition Function
We can perform the four basic operations between two functions: (Kl . 13%)
Addition: «P r (x7: J; (K) lr 30)
Subtraction: (43004) : «COO 3650
Multiplication:
Division: f
( '
Ex
Math 138: Section 1.9
Inverse Functions
Defn: A function f'1(x), read as "f inverse of x is called the inverse, off(x) ifand
only if (f°f1)(x) = x AND (flcfXx) = x
The domain off(x) is the gang ('2 off1(x), and the Yang f» of
f(x) is the LLDMQUB off1(x),i
Final Exam Practice Solutions
Math 114
Spring 2013
Corwin
1. The bar graph below shows data about the birthdays of students in Mrs. Jones class.
How many students have birthdays before April 1?
3 + 4 + 2 = 9.
2. Jane has made the following table to comput
1
Chapter 8: Congruences
Practice HW p. 62 # 1, 2, 3, 5, 6, Additional Web Exercises
In this section, we look at the fundamental concept of modular arithmetic, which is used
in a variety of applications in number theory.
Modular Arithmetic
Definition: Giv
1
Supplemental Section: Factoring Integers
Practice HW Exercises 14 at the end of these notes
In this section, we examine techniques to obtain the prime factorization.
Square Root Test for Prime Factorization
Fact: A number m is prime if it has no prime
1
Chapter 19: Primality Testing and Carmichael Numbers
Practice HW p. 127 # 1, 2, 3a, 4
Recall Fermats Little Theorem says that if p is a prime number and a is an integer where
p / a , then

a p 1 1 (mod p )
If we multiply both sides by a, we obtain the
1
Chapter 15: Mersenne Primes and Perfect Numbers
Practice HW p. 108 # 1, 2, 3a, 3b
In this section, we examine how to use Mersenne primes to generate perfect numbers. We
start with the following definition.
Definition 15.1: A number n is a perfect number
1
Chapter 5: Divisibility and the Greatest Common Divisor
Practice HW p. 34 # 1, 4, Additional Web Exercises
The Greatest Common Divisor of Two Positive Integers
Definition: The greatest common divisor of two positive integers a and b , denoted as
gcd(a,
1
Chapter 10: Congruences, Powers, and Eulers Theorem
Practice HW p. 74 # 2, 3a, Additional Web Exercises
In this section, we state and prove Eulers Theorem and look at its benefits.
EulerPhi Function
Given an integer m, the EulerPhi function, denoted b
1
Chapter 14: Mersenne Primes
Practice HW p. 99 # 1, 3
We want to consider under what conditions the quantity a n 1 is prime. We establish
these conditions with the following facts.
Fact 1: a n 1 is even if a is odd.
Proof:
Fact 2: ( a 1)  ( a n 1 )
Proo