MATH 111
Homework 11
1. Determine which of the following functions are injective, surjective, or bijective. Prove your answer!
(a) f : R R where f (x) = 7x 5
2x2 1
x2 + 1
(c) f : Z Z Z Z where f (m, n) = (m + n, m n).
(b) f : R R where f (x) =
Solution :
PRIME, MODULAR
ARITHMETIC, AND
By: Tessa Xie
&
Meiyi Shi
OBJECTIVES
Examine Primes In Term Of Additive Properties & Modular
Arithmetic
To Prove There Are Infinitely Many Primes
To Prove There Are Infinitely Many Primes of The Form 4n+2
To Prove There
Section 1.3
1
Predicate Logic
Section 1.3:
1.3: Predicate Logic
Purpose of Section:
Section: To introduce predicate logic (or firstfirst-order logic)
logic which
the language of mathematics. We see how predicate logic extends the
language of sentential ca
1) Negate the following statement: given x (0, ), we have that x > y > 0 for some
y (0, ). [As usual, your answer should contain no nots.]
Solution. There exists x (0, ), such that for all y (0, ), x y or y 0.
2) Let A, B, C be subsets of U . [You can use
Math 216 First Exam
February 20, 2017
Your Name:
EXAM SOLUTIONS
Your Instructor:
1. Do not open this exam until you are told to do so.
2. This exam has 6 pages including this cover. There are 5 problems. Note that the problems
are not of equal difficulty,
Math 4426: Spring 2016
1. (24 pts):
Exam 1: Solutions
Page 1 of 5
Suppose you deal a 7-card hand from a standard 52-card deck.
(a) What is the probability your hand contains a 4 of a kind and a 3 of a kind (i.e., your 7
cards have values x, x, x, x, y, y,
Math 4426: Spring 2016
1. (20 pts):
Exam 3: Solutions
Page 1 of 4
Let X and Y be independent random variables with Var(X) = 2 and Var(Y ) = 3.
(a) Find Var(3X Y + 1).
(b) Find Cov(X Y, X + Y ).
Solution:
(a) Since X and Y are independent, we have that
Var
Math 4426: Spring 2016
Exam 2: Solutions
Page 1 of 5
1. (16 pts): A certain winery sells cases of wine that contain 12 bottles per case. Data suggests
that 7% of the bottles of wine produced by this winery are bad bottles.
(a) Let X be the random variable
Math 4426
Page 1 of 5
Fall 2016
Final Exam Review Solutions
Problem 1:
(a) Let X be the number of shots needed for Bernie to hit the target for the first time.
Then X Geom(0.8). Hence, the desired probability is
P(X 2) = P(X = 1) + P(X = 2) = 0.8 + (1 0.8
Math 4426
Page 1 of 2
Fall 2016
Final Exam Review Problems
1. Bernie Oullie is practicing his archery. Specifically, he is practicing hitting a target
from 100 meters. On any one attempt, he has a 80% chance of hitting the target and
the outcomes of previ
MATH 4427 Problem Set 5
Due Wednesday, December 5, 2016
1. 9.2.2 Assume normality.
Letting weight loss under Atkins represent the first population and that under Zone the
second, we have x = 4.7, sX = 7.05, n = 77, and X = mean weight loss on Atkins diet.
Math 4426
Page 1 of 5
Fall 2016
Homework # 10 Solutions
Due date: Monday 12/5 in class
Problem 7.75:
From the moment generating functions of X and Y defined in the problem, we can use our
distribution chart to determine that X Poi(2) and Y Bin(10, 3/4). N
Math 4426
Page 1 of 7
Fall 2016
Homework # 9 Solutions
Due date: Monday 11/21 in class
Problem 6.38:
Let X be (discrete) uniformly distributed on the set cfw_1, 2, 3, 4, 5. Then, given that X = x,
suppose that Y is uniformly distributed on the set cfw_1,
Midterm 2 Topics
This midterm will cover all the material that we have gone over in class from
the following sections:
Ch 3.3: Operations on sets (plus Cartesian products),
Ch 3.4: Functions,
Ch 3.5: Image and preimage,
Ch 4.1: Equivalence relations a
Problem Set Three Solutions
Problem 1. Suppose I build a jungle gym in the shape of a ` m n grid.
That is, the jungle gym measures ` feet from north-to-south, m feet from eastto-west, and n feet from bottom-to-top. How many paths are there from the
south-
Problem Set One Solutions
Problem 1. Prove that if x Q, y
/ Q and x 6= 0 6= y, then xy
/ Q.
Solution. To get a contradiction, assume that xy Q. Then we can write
xy = a/b, where a Z and b Z+ . Since x Q and x 6= 0, we can also write
x = c/d, where c Z+
Problem Set Two Solutions
Problem 1 (Ex. 2.19). Prove that 5 2n 3n for all integers n 4.
Solution. Let P (n) be the statement that 5 2n 3n . We first check the base
case, n = 4. P (4) says that 5 24 34 , which is true since 5 24 = 80 and
34 = 81.
Now, sup
Midterm 1 Topics
This midterm will test your knowledge of the material we have gone over in
class from the beginning of the semester until the end of Chapter II of the Course
Notes. That is, everything up to and including the proof of the Fundamental
Theo
Problem Set Four Solutions
Problem 1 (Ex. 1.4). Suppose a, b, c Z. For each of the following claims,
give either a proof or a counterexample.
(a) If a | b and b | c then a | c.
(b) If a | bc then either a | b or a | c.
(c) If a | c and b | c then ab | c.
Problem Set Five Solutions
Problem 1 (Ex. 1.5). Suppose b, c Z+ are relatively prime and a is a divisor
of b + c. Prove that
gcd(a, b) = 1 = gcd(a, c).
Solution. Well show that gcd(a, b) = 1. The proof that gcd(a, c) = 1 is
basically the same. Suppose k i
Problem Set Eight
Your Name Here
November 8, 2013.
Read the definition of partition (Def. 1.10) and the discussion that follows on
pg. 35 of the course notes.
Problem 1 (Ex. 1.18). Let f : X
Y be a function, and for every y Y
let Ay = f 1 (cfw_y). The s
Problem Set Seven
Your Name Here
November 1, 2013.
For all but the last the problem below, we assume that f is a function from
X to Y .
Problem 1 (Ex. 5.5). Suppose that f is surjective. Prove that every A X
satisfies
Y r f (A) f (X r A).
Show by example
Problem Set Solutions
Problem 1 (Ex. 4.10). For each of the following functions either compute the
inverse or show that no inverse exists.
(a) f : R
R defined by f (x) = (5x 2)/12
(b) f : R
R defined by f (x) = x2
(c) f : Z
Z defined by f (n) = 2n.
MATH 4427
Midterm 2 solutions November 11, 2016
Show all your work
1. [12 pts] The director of a national jobs training program claims that the mean age of
participants is less than 20. In a random sample of 13 participants, the mean age was 19.3
years. A
Math 4426: Fall 2016
Exam 2: Solutions
Page 1 of 5
1. (18 pts): Suppose a box contains 100 computer chips, of which exactly 5 are defective and
the remaining 95 are in good working order. In each of the following parts, determine the
probability mass func
Findthevolumeofthesolidgeneratedbyrevolvingtheregionboundedbythegivenlinesandcurvesaboutthe x-axis.
21) y=x,y=0,x=1,x=5
21)
124
4
C)
D) 13
A) 12
B)
3
3
Answer: C
22) y= x,y=0,x=0,x=9
81
A)
2
22)
B) 9
9
C)
2
D) 27
B) 9
243
C)
4
243
D)
5
Answer: A
23) y=x