IN ORDER TO IMPLEMENT A SET OF RULES / TUTORIALOUTLET DOT COM

AC 202 Solve each trigonometric equation
FOR MORE CLASSES VISIT
www.tutorialoutlet.com
Solve each trigonometric equation in the interval [0,2n) by first
squaring both sides. fleas x=1+sin x Select the correct choice below
and, if necessary, fill in the answer box to complete your choice. O A-
The solution set is .
(Simplify your answer. Use a comma to separate answers as needed.
Type an exact answer, using 1: as needed. Use integers or fractions
for any numbers in the expression.) 0 B. There is no solution on this
interval.
Find all solutions of the equation in the interval [0,21r). tan2x=3
Select the correct choice below and, if necessary, fill in the answer
box to complete your choice. O A- The solution set is .
Solve the equation. 2sln29+sln9-1=0 What is the solution in the
interval 0 S 9 < 21:? Select the correct choice and fill in any answer
boxes in your choice below. O A- The solution set
Use trigonometric identities to solve the equation in the interval
[0,21r).
5cose= -54/§sine Select the correct choice below and, if necessary, fill
in the answer box to complete your choice. O A- The solution set is
Use trigonometric identities to solve the equation in the interval [0,
21:). sin 2x— cos 2x=1 Select the correct choice and fill in any
answer boxes in your choice below. 0A— x={D} (Type your answer
in radians. Type an exact answer, using 1: as needed. Use integers or
fractions for any numbers in the expression. Use a comma to separate
answers as needed.)
Find all solutions of the equation in the interval [0, 21:).
(tanx+1)(2 sinx—1)=0
Select the correct choice below and, if necessary, fill in the answer
box to complete your choice. 0A- x={j} (Simplify your answer. Type

an exact answer, using 1r, as needed. Type your answer in radians.
Use integers or fractions for any numbers in the expression. Use a
comma to separate answers as needed.)
Find all solutions of the equation in the interval [0, 21:). 8 sln 29 = 2
Select the correct choice and ﬁll in any answer boxes in your choice
below. 0.. e={C} (Simplify your answer. Type an exact answer, using
1|: as needed. Type your answer in radians. Use integers or fractions
for any numbers in the expression. Use a comma to separate answers
as needed.)
------------------------------------------------------------------------------------
BTE 200 The following formula gives the distance between
two points
The following formula gives the distance between two points, (x1, y1)
and (x2, y2) in the
Cartesian plane: (1‗2 — 1171)2 + (3/2 — 902 Given the center and a
point on the circle, you can use this formula to find the radius of the
circle.
Write a program that prompts the user to enter the center and a point
on the circle. The program
should then output the circle‘s radius, diameter, circumference, and
area. Your program must
have at least the following functions: a. distance: This function takes
as its parameters four numbers that represent two points in the
plane and returns the distance between them. b. radius: This function
takes as its parameters four numbers that represent the center and a
point on the circle, calls the function distance to find the radius of the
circle, and returns the
circle‘s radius. c. circumference: This function takes as its parameter
a number that represents the radius of the
circle and returns the circles circumference. (If r is the radius, the

circumference is 2 nr.) d. area: This function takes as its parameter a
number that represents the radius of the circle and
returns the circle‘s area. (If r is the radius, the area is 11r2.) Assume
that n = 3.1416.
------------------------------------------------------------------------------------
Chapter 8 Hypothesis Testing and Types of Errors
Business Statistics
Chapter 8
Hypothesis Testing and Types of Errors Business Statistics January 4,
2017 ( Business Statistics) Chapter 8 January 4, 2017 1 / 48 1
Introduction to Hypothesis Testing 2 Hypothesis Testing for p 3
Hypothesis Testing for µ 4 Independent Samples: Hypothesis Testing
for µ1 − µ2 5 Dependent Samples: Hypothesis Testing for µdiff 6
Type I and Type II Errors ( Business Statistics) Chapter 8 January 4,
2017 2 / 48 Introduction to Hypothesis Testing Introduction to
Hypothesis Testing As we saw in the previous chapter, confidence
intervals estimate a
population mean or proportion by giving a range it likely falls in
using
a random sample of data.
Hypothesis testing has the same goal as a confidence interval,
informing us about a population mean or proportion, but with a
different approach.
A hypothesis test allows us to come to a conclusion about the mean
or proportion. ( Business Statistics) Chapter 8 January 4, 2017 3 / 48
Introduction to Hypothesis Testing Introduction to Hypothesis Testing
Let‘s look at some examples of when hypothesis testing would be
used.
1 After the recession, the fees charged by banks for late credit card
payments have increased. 2 The industry that promotes compact
fluorescent light bulbs claims the

bulbs use 75% less energy and last 10 times longer than incandescent
bulbs. 3 A Wall Street Journal article, from June 2010, titled ‗Does
the
Internet Make You Smarter or Dumber?‘ posed the possibility that
online activities turn us into shallow thinkers. Each of these examples
could be stated as a pair of opposing claims or
hypotheses. ( Business Statistics) Chapter 8 January 4, 2017 4 / 48
Introduction to Hypothesis Testing Null vs. Alternative
Looking at the three examples above, state the null and alternative
hypotheses.
General form
H0 vs. H1
1 H0 : The fees charged by banks for late credit card payments have
remained the same or decreased (not increased) after the recession.
H1 : The fees charged by banks for late credit card payments
increased after the recession. 2 H0 : Fluorescent light bulbs do not use
75% less energy and last 10
times longer than incandescent bulbs.
H1 : Fluorescent light bulbs use 75% less energy and last 10 times
longer than incandescent bulbs. 3 H0 : Online activities do not turn us
into shallow thinkers.
H1 : Online activities turn us into shallow thinkers.
( Business Statistics) Chapter 8 January 4, 2017 5 / 48 Introduction to
Hypothesis Testing Null vs. Alternative Hypothesis testing typically
begins with some claim (or belief) about
a particular parameter of a population.
Our initial assumption is known as the null hypothesis, denoted H0
(‗H naught‘). This typically is hinting that nothing is happening, the
status quo, no relationship, no difference, etc.
The null hypothesis is believed to be true unless there is
overwhelming evidence not to.
We use the sample data to see if the alternative hypothesis, denoted
Ha or H1 , is true.
The alternative hypothesis states that something is going on: a
difference, increase, decrease or relationship exists.
Typically, H1 is what the researchers hope to show. ( Business
Statistics) Chapter 8 January 4, 2017 6 / 48 Introduction to

Hypothesis Testing Hypothesis Testing Let‘s consider the American
justice system of ‗innocent until proven
guilty‘.
If we were to state this as a hypothesis, this would be:
H0 : The defendant is innocent.
H1 : The defendant is guilty. We make the initial assumption that the
defendant is innocent and
evidence at the trial (think sample data) either shows the claim of
guilt or it doesn‘t.
The evidence (sample data) will lead us to a verdict (conclusion). (
Business Statistics) Chapter 8 January 4, 2017 7 / 48 Introduction to
Hypothesis Testing Hypothesis Testing A hypothesis test is a five step
process.
1
2
3
4 5 State the hypotheses.
Calculate the test statistic (which is essentially a z-score)
Determine the p-value (we will discuss this in a moment).
Make a decision (where we side with either H0 or H1 ). To do this,
we
will need a significance level.
State a conclusion in terms of the problem. ( Business Statistics)
Chapter 8 January 4, 2017 8 / 48 Introduction to Hypothesis Testing
The p-value and Significance Level The p-value is the probability of
obtaining a test statistic as extreme
or more extreme than the one observed when we assume that H0 is
true.
We then compare the p-value to a significance level, denoted α, which
is similar to the confidence level for a confidence interval.
Typically, we set α = 0.05, which is equivalent to the concept of 95%
confidence intervals having a 95% success rate and a 5% failure rate.
( Business Statistics) Chapter 8 January 4, 2017 9 / 48 Introduction to
Hypothesis Testing The p-value and Significance Level In order to
make a conclusion about which hypothesis is more likely to
be correct, we have the following decision rule.
If p-value ≤ α, we declare the result significant and ‗reject H0 ‘.

If p-value > α, we ‗fail to reject H0 ‘, The data did not provide
significant evidence to reject H0 . Please note that we never ‗accept
H0 ‘.
The problem of ‗accepting‘ is that this wording makes it seem that we
are convinced that the null hypothesis is true. ( Business Statistics)
Chapter 8 January 4, 2017 10 / 48 Introduction to Hypothesis Testing
The p-value and Significance Level Hypothesis testing is meant to
reject the null hypothesis when the
evidence is convincingly against it.
If we collect only a small sample, we may not see convincing
evidence
for the alternative hypothesis because the sampling error is so large.
For instance, if we observed three births and observed two boys, we
would not be willing to accept the hypothesis ‗2/3 of all births are
boys‘ even though the data doesn‘t provide evidence that it is false. (
Business Statistics) Chapter 8 January 4, 2017 11 / 48 Hypothesis
Testing for p Hypothesis Testing for p The pharmaceutical company
is claiming a hypothesis, or wanting to
show, that fewer than 20% of the patients who use a particular
medication experience side effects.
In a clinical trial with 400 patients, they find 68 patients experienced
side effects.
Initially we would assume that 20% or more of patients who use a
particular medication experience side effects. ( Business Statistics)
Chapter 8 January 4, 2017 12 / 48 Hypothesis Testing for p
Hypothesis Testing for p This can be written as:
H0 : 20% or more of the patients who use a particular medication
experience side effects.
H1 : Fewer than 20% of the patients who use a particular medication
experience side effects. We can also summarize this as:
H0 : p ≥ 0.20
H1 : p < 0.20
where p is the true proportion of patients who use a particular
medication that experience side effects. ( Business Statistics) Chapter
8 January 4, 2017 13 / 48 Hypothesis Testing for p Hypothesis
Testing for p Once we have the hypotheses stated, we then need to
calculate the

test statistic (z-score in this case).
From our sample we see that of 400 patients, 68 patients experienced
side effects.
This is equivalent to 17% (or pˆ = 68
400 = 0.17). From the sample we are already seeing some evidence to
support the
alternative (0.17 < 0.20), but is this significant? ( Business
Statistics) Chapter 8 January 4, 2017 14 / 48 Hypothesis Testing for p
Hypothesis Testing for p Recall that the z-score for a single
proportion is
−p
Z = pˆSE
q
where SE = p(1−p)
n
Note that we are using the standard error for p in the denominator
since we are assuming H0 is true. ( Business Statistics) Chapter 8
January 4, 2017 15 / 48 Hypothesis Testing for p Hypothesis Testing
for p 0.17−0.20
0.02 0.3
0.1 Z = −1.5 0.2 Probability Z= 0.4 Using the information provided
in the problem, we obtain
q
SE = 0.2(1−0.2)
= 0.02
400 0.0 Once we obtain the test
statistics, the next step is to
calculate the p-value. −4 −2 0 2 4 Z < −1.5 Remember that the p-
value is
the probability of obtaining a
test statistic as extreme or more
extreme. In this example we are
looking for a Z = −1.5 or
smaller (i.e.
P[Z ≤ −1.5]).
P[Z ≤ −1.5] = 0.0668 ( Business Statistics) Chapter 8 January 4, 2017
16 / 48 Hypothesis Testing for p One-tailed or Two-tailed Before we

continue with the example, we come to an important
distinction when calculating p-values.
In this example we are calculating the p-value for the alternative
hypothesis H1 : p < 0.20.
What we will see when shading the area we wished to calculate is that
we are looking for the area less than Z = −1.5.
There are three possible alternative hypotheses, those being
H1 : p < p0
H1 : p > p0
H1 : p 6= p0
where p0 is the null value. ( Business Statistics) Chapter 8 January 4,
2017 17 / 48 Hypothesis Testing for p One-tailed or Two-tailed The
first two types of alternative hypotheses are considered
one-tailed.
------------------------------------------------------------------------------------
CMPS 12A write a C program that operates in the same
way
write a C
program that operates in the same way, i.e. same prompts, same input
and same output. However, the
requirements for this program are relaxed somewhat from those in
pa3 in that it is not necessary to filter out
all types of bad input.
Design your program to respond to string inputs by printing
Please enter a positive integer: as specified in pa3, then scan for
another integer. Respond similarly to integer input that is negative or
zero. It is not necessary to react to double inputs according to the pa3
specifications however. Thus you
may assume that things like ―25.78‖ will not be used as input to your
program. Everything you need to do
this was explained in the lab7 project description. In particular,

review the explanation of the scanf()
function given in that document before you begin this program. A
sample session follows.
% GCD
Enter a positive integer: sldkfj
Please enter a positive integer: -56
Please enter a positive integer: 56
Enter another positive integer: sldkjfdlk
Please enter a positive integer: -25
Please enter a positive integer: 25
The GCD of 56 and 25 is 1
% Recall the CheckInput sequence of examples in Java whose
purpose was to learn how to filter input from
standard input. A similar sequence of examples will be posted under
Examples/lab8 on the class
webpage. Study these carefully to learn how to read and discard non-
numeric string input.
Call your program GCD.c and write a Makefile that creates an
executable file called GCD. Include a clean
utility with the Makefile that deletes the executable. Submit both files
to the assignment name lab8. 1
------------------------------------------------------------------------------------
Convert the polar coordinates
1- Convert the polar coordinates (-3, 135º) into rectangular
coordinates. Round the rectangular coordinates to the nearest
hundredth.
a) (-2.12, -2.12) b) (-2.12, 2.12) c) (2.12, 2.12) d) (2.12, -2.12)
2-The letters x and y represent rectangular coordinates. Write the
following equation using polar coordinates (r, 0).

x^2 + y^2 -4x = 0
a) r=4sin0 b) r=4cos0 c) rcos^2 0 =4sin0 d) rsin^2 0= 4cos0
3- The letter x and y represents rectangular coordinates. Write the
following equation using polar coordinates (r, 0)
x^2+4y^2=4
a) cos^2 0 + 4sin^2 0 = 4r b) 4cos^2 0 + sin^2 0 =4r c) r^2(cos^2
0 + 4sin^2 0) = 4 d) r^2(4cos^2 0 + sin^2 0) = 4
4) The letters r and 0 represent polar coordinates. Write the following
equation using rectangular coordinates (x, y).
r=10sin0
a) (x+y)^2=10x b) (x+y)^2=10y c) x^2+y^2=10y d) x^2+y^2=10x
------------------------------------------------------------------------------------
CS 2400 FUNCTIONS In addition to function
FUNCTIONS In addition to function main, a program source module
may contain one or more other functions. The execution of the
program always starts with function main. The statements of a
function (other than function main) are executed only if that function
is called in
function main or any other function that is called in function main.
After the statements of a function are executed, the next statement to
be executed is the one that
follows the statement in which that function is called. Example
The statements of the program in figure F1 are executed in the
following order: 23, 24, 26, 15, 16,
27, 28, 31, 32, 34, 15, 16, 35, 36, and 37. There are two types of
functions in the C/C++ programming language: Functions that do not

return a value: void functions, and Functions that return a value. You
can also write a function with or without parameters: An example of a
void function without parameters is provided in figure F1, line 13 to
line 17. An example of a void function with parameters is provided in
figure F2, line 8 to line 12. Defining and Calling a void Function
without Parameters You define a void function without parameters as
follows: void <function-name>( )
{
<Body-of-the-function>
}
<function-name> is the name of the function void <function-
name>( ) is the function header.
It may also be specified as follows: <Body-of-the-function>
©2011 Gilbert Ndjatou void <function-name>( void ) is the
body of the function: It consists of one or more statements that are
executed each time the function is called.
Page 169 You call a void function without parameters by using the
following statement:
<function-name> ( ); The program in figure F1 illustrates the
definition and calls of a void function without parameters. Figure F1
Defining and Calling a void Function without Parameters 1. /*-----
Program to compute the area and the perimeter of rectangles ------*/
2.
3. #include
<iostream>
4. using namespace std;
5.
6. double len,
// to hold the length of the rectangle
7.
width
// to hold the width of the rectangle
8.
area,
// to hold the area
9.
peri;

// to hold the perimeter
10.
11.
/*------------------------function computeAreaPeri1 ---------------------*/
12.
/*------- compute the area and the perimeter of a rectangle -----------*/
13.
void computeAreaPeri1( void )
14.
{
15.
area = len * width;
16.
peri = 2 * ( len + width );
17.
}
18.
19.
int main ()
20.
{
21.
/*-------compute and print the area and the perimeter of a rectangle
with
length 20 and width 8 ---------------------------------------------*/
22.
23.
len = 20;
24.
width = 8;
25.
26.
computeAreaPeri1( );
27.
cout << endl << ―the area of the rectangle is:t‖
<< area;
28.

cout << endl << ―Its perimeter is:t‖
<<
peri;
29.
30.
/*-------read the length and the width of a rectangle and compute and
print its area and perimeter ------------------------------------- */
31.
cout << endl << ―enter the length and the width of the
rectangle:t‖;
32.
cin
>> len
>> width;
33.
34.
computeAreaPeri1( );
35.
cout << endl << ―the area of the rectangle is:t‖
<< area;
36.
cout << endl << ―Its perimeter is:t‖
<<
peri;
37.
return ( 0 );
}
© 2011 Gilbert Ndjatou Page 170 Global Variables and Local
Variables A global variable is a variable that is defined outside of the
body of any function.
Examples of global variables are variables len, width, area, and peri
defined in figure F1, line 6 to
line 9. A global variable can be accessed in the body of any function
that appears after its definition. Example
In the source module in figure F1,
len
is accessed in the body of function computeAreaPeri1 in lines 15 and

16
and in the body of function main in lines 23 and 32. A function can
use global variables to share information with the calling function.
Example
In the source module in figure F1, Function main stores 20 into
variable len in line 23, and 8 into variable width in line 24 before it
calls function computeAreaPeri1 in line 26. In line 15, function
computeAreaPeri1 retrieves 20 from variable len and 8 from variable
width,
multiplies 20 by 8, and stores the result, 160.0 in variable area. In line
16, function computeAreaPeri1 retrieves 20 from variable len and 8
from variable width,
computes 2 * (20 + 8), and stores the result, 56.0 in global variable
peri. In line 27, function main retrieves 160.0 from global variable
area and prints it. In line 28, function main retrieves 56.0 from global
variable peri and prints it. A variable that is defined in the body of a
function is a local variable of that function. A local variable can only
be accessed in the body of the function in which it is defined.
Exercise F1*
Execute the following program and show its output for the input value
7: ©2011 Gilbert Ndjatou Page 171 #include <iostream>
using namespace std;
int gnum1, gnum2;
void funct(void)
{
int num = gnum1 + 10;
gnum1 + = num;
gnum2 = 2 * gnum1 + 5;
}
int main( )
{
gnum1 = 15;
funct( );
cout << endl << ―gnum1=‖ << gnum1 <<
―tgnum2=‖ << gnum2;
cin >> gnum1;
funct( );

return 0;
} Exercise F2*
Write a void function without parameters computeAreaPeri that
computes the area and the perimeter of
a circle which are output in function main.
a. Function main first calls function computeAreaPeri to compute the
area and the perimeter of the
circle with radius 5.43.
b. It then reads the radius of a circle and then calls function
computeAreaPeri again to compute the
area and the perimeter of this circle.
You must first determine and define the global variables of this
program; then write function
computeAreaPeri, and finally write function main. Exercise F3
Execute the following program and show its output for the input value
15:
#include <iostream>
using namespace std;
int gnum1, gnum2; ©2011 Gilbert Ndjatou Page 172 void funct(void)
{
int num = 2 * gnum1 + 5;
gnum2 = gnum1 + num;
gnum1 = gnum2 + 10;
}
int main( )
{
gnum1 = 25;
funct( );
cin >> gnum1;
funct( );
return 0;

} Exercise F4
Write a void function without parameters computeTaxNet that uses
the gross pay of an individual to
compute his tax deduction and net pay that are printed in function
main. The tax deduction is computed
as follows: if the gross pay is greater than or equal to $1000.00, then
the tax deduction is 25% of the
gross pay; otherwise, it is 18% of the gross pay. The net pay is the
gross pay minus the tax deduction.
a. Function main first calls function computeTaxNet to compute the
tax deduction and the net pay for
the gross pay $1250.
b. It then reads the gross pay of an individual and then calls function
computeTaxNet to compute his
tax deduction and net pay.
c. You must first determine and define the global variables of this
program; then write function
------------------------------------------------------------------------------------
Determine whether they form a partition for the set of
integers
For each the following groups of sets, determine whether they form
a partition for the set of integers. Explain your answer.
a. A1 = {n Z: n > 0}
A2 = {n Z: n < 0}
A1 contains all integers greater than 0.
A2 contains all integers less than 0.

0 is not covered in both cases.
A1 ∩ A2 = ∅ ; however, A 1∪A 2 ≠ Z , as 0 is missing. Therefore, it
is not
a partition of the set of all integers.
.b. B1 = {n Z : n = 2k, for some integer k}
B2 = {n Z : n = 2k + 1, for some integer k}
B3 = {n Z : n = 3k, for some integer k}
B1 is the set of all even integers, B2 is the set of all odd integers, and
B3
is the set of all integers divisible evenly by 3. E.g. B3 = {3, 9, 12, 15,
18…}
This is a partition of Z since B1 U B2 U B3 ¿ Z.
2. (10 pts) Define f: Z → Z by the rule f(x) = 6x + 1, for all integers x.
a. Is f onto?
No. For a function to be onto the codomain must equal the range. A
counter example would be f(x)=y and solve for x. We find that x = y-
1/6.
Let y=0 and x = -1/6. This means that, in order to get 0 (an integer) as
the
resulting value for f(x), we have to input -1/6 into the function. -1/6 is
not
an integer; therefore, the function is not onto.
b. Is f one-to-one?
Yes, f is one-to-one. For every x there will be a different f(x) and for
every

f(x) there will be a different x.
c. Is it a one-to-one correspondence?
For a function to be a one-to-one correspondence it must be both one-
to-one and
onto. It must have a codomain equal to the range, and each element of
the domain
must map to only a single element in the range so based on exercise a
we can see that
is not onto therefore is not one-to-one correspondence. d. Find the
range of f
Range = {n ∈ Z | 6n+1 ∈
= {…, -11, -5, 1, 13, 19, …} Z} 3. (10 pts) f: R → R and g: R → R
are defined by the rules:
f(x) = x2 + 2 ∀
g(y) = 2y + 3 ∀ x ϵ R y ϵ R Find f ◦ g and g ◦ f
2
2
f ◦ g = f ( g ( y ) )=(2 y+ 3) +2=4 y +12 y +11
2
f ◦ g = 4 y +12 y +11 g ◦ f = g(f(x)) = g(x2 + 2)
=> g(x2 + 2) = 2(x2 + 2) + 3
=> 2x2 + 4 + 3 = 2x2 + 7
g ◦ f = 2x2 + 7

4. (10 pts) Determine whether the following binary relations are
reflexive,
symmetric, antisymmetric and transitive:
a. x R y ⇔ xy ≥ 0 ∀ x, y ϵ R Reflexive - Any relation to be reflexive,
(x,x) should belong to R.
If we consider any value of x then x*x will always be an positive
value >0. For
example X=2, Y=2 2*2 > = 0 or X= -4 Y= -4, -4*-4> = 0
therefore we can say R is
reflexive. Symmetric - any relation to be symmetric, (x,y) should
belong to R and (y,x)
should also belong to R. here for any value of x and y if (x,y) belongs
to R i.e,
x*y>=0 then y*x will also be > = 0 thus (y,x) will also belong
to R. It is also
symmetric. Not antisymmetric because it is symmetric. Transitive -
any relation to be transitive, must hold if (x,y) and (y,z) belongs to R
then (x,z) should belong to R. When x*y>=0 and y*z>=0 the
we can say x*z will
also be >=0, thus (x,z) belongs to R. Is∀an
relation.
x equivalence
,
x> 0 [ x ] = {∀ y∨ y> 0 } ,
x< 0 [ x ] = { ∀ y| y< 0 } ,

x=0 [ x ] = R b. x R y ⇔ x > y ∀ x, y ϵ R Not reflexive: A
counterexample to prove is not would be x=y, x=4; therefore, x
should be greater than y, but since 4 is not greater than 4, this
relationship is not
reflexive. Not Symmetric - any relation to be symmetric, (x,y) should
belong to R and (y,x)
should also belong to R. For any value of x and y if (x,y) belongs to R
i.e, x>y
then y>x is not possible so we can say that R is not symmetric
There is no x, y pairs that relate back to each other. E.g. (x, y) is
found, but not
(y, x) for all x, y in R. Therefore, it is antisymmetric. Transitive - x, y,
z are related. If x > y, and y > z, then x > z. Is not an
equivalence relation, nor partial order.
c. x R y ⇔ |x| = |y| ∀ x, y ϵ R Reflexive - Any relation to be reflexive,
(x,x) should belong to R. If we consider
any value of x then |x|=|x| will hold. R is reflexive Symmetric - It is
symmetric was for all (x, y) there is a corresponding (y, x) pair.
E.g. (-1, 1), (1, -1). Because it is symmetric cannot be antisymmetric.
Not transitive since no number is related to each other. Is not an
equivalence relation, nor partial order. 5. (10 pts) Determine whether
the following pair of statements are logically
equivalent. Justify your answer using a truth table.
p → (q → r) and p ∧ q → r p ∧ q → r p (q r)
p Q r T
T
T

T
F
F
F
F T
F
T
F
T
F
T
F T
T
F
F
T
T
F
F (q r)
T
F

T
T
T
F
T
T p p (q r)
T
F
T
T
T
T
T
T Statements are logically equivalent. T
T
T
T
F
F
F q T
T

F
F
T
T
F T
F
T
F
T
F
T r T
T
F
F
F
F
F p
∧ p q q ∧ → r
T
F
T

T
T
T
T 6. (10 pts) Prove or disprove the following statement:
∀ n, m ∈ Z, If n is even and m is odd, then n + m is odd
Then write the negation of this statement and prove or disprove it.
n + m is not odd
Given that n is even and m is odd, is always going to be odd;
therefore, the negation of
the previous statement is false. 7. (10 pts) Prove the following by
induction:
n
n+1
3 n2−n
3
i
–
2=
∑
=> ∑ ( 3 i – 2 ) +(3 n+1)
2
i=1

i=1
2 3 (n+1) −(n+ 1)
3 n 2−n
+(3 ( n+1 ) −2)=
2
2 2 3n^2 – n / 2 + (3n+1) = 3 n +5 n+ 2
2 8. (10 pts) Use the permutation formula to calculate the number
permutations of
the set {V, W, X, Y, Z} taken three at a time. Also list these
permutations. 5P3 = 5!
5!
= =60
( 5−3 ) ! 2! permutations {V,W,X} {V,W,Y} {V,W,Z} {V,X,W}
{V,X,Y} {V,X,Z} {V,Y,W} {V,Y,X} {V,Y,Z} {V,Z,W} {V,Z,X}
{V,Z,Y} {W,V,X} {W,V,Y} {W,V,Z} {W,X,V} {W,X,Y} {W,X,Z}
{W,Y,V} {W,Y,X} {W,Y,Z}
{W,Z,V} {W,Z,X} {W,Z,Y} {X,V,W} {X,V,Y} {X,V,Z} {X,W,V}
{X,W,Y} {X,W,Z} {X,Y,V}
{X,Y,W} {X,Y,Z} {X,Z,V} {X,Z,W} {X,Z,Y} {Y,V,W} {Y,V,X}
{Y,V,Z} {Y,W,V} {Y,W,X} {Y,W,Z}
{Y,X,V} {Y,X,W} {Y,X,Z} {Y,Z,V} {Y,Z,W} {Y,Z,X} {Z,V,W}
{Z,V,X} {Z,V,Y} {Z,W,V} {Z,W,X}
{Z,W,Y} {Z,X,V} {Z,X,W} {Z,X,Y} {Z,Y,V} {Z,Y,W} {Z,Y,X} 9.
(10 pts) Translate the following English sentences into statements of
predicate

calculus that contain double quantifiers and explain whether it is a
true
statement.
a. Every rational number is the reciprocal of some other rational
number.
P ( x ) ∙ N ( x )=1
( ∀ P ( x ) ϵ θ ) (∃ N ( x ) ϵ θ) ¿
statement is true
Reciprocal is inversion of rational number. Dividing would equal 1.
b. Some real number is bigger than all negative integers.
x is real numbers
( ∃ x ∈ R ) , x> y
y is negative integers
statement is true.
x = 2 and y = -4
2>4
10. (10 pts) Consider the following graph: In each case, answer the
question and then write the rationale for your answer.
a. Is this graph connected? Yes, no corners are separated from the rest
of the graph
b. Is this a simple graph?
Yes, there are no multi edges. There are no loops nor parallel edges.
c. Does this graph contain any cycles?

It is possible if we start in the middle traveling to the left and then
come
back to the starting point.
d. Does this graph contain an Euler cycle?
This cycle requires that all edges be used in a path that starts and
stops at
the same vertex.
Is this graph a tree?
No. It does not have any open ends or a root vertex.
------------------------------------------------------------------------------------
ECON 215 Students in an introductory psychology course
Chapter 5 Students in an introductory psychology course take five
quizzes and two exams throughout the
semester. Each year, approximately 500 students take the course. The
first quiz has 10 multiplechoice questions where each question has
four choices with only one correct answer. The
passing score on the quiz is 80%.
a. (4 pts) If a student must resort to pure guessing on each question
(selects one of the four
answer choices randomly on each question),
What is the probability the student will pass the quiz? __________
b. (3 pts) If a student knows the material well enough to be able to
eliminate 2 incorrect choices
on each question but selects the answer randomly from the 2
remaining choices,
c. (3 pts) If a student knows the correct answer to five of the ten

questions, but must resort to
pure guessing on the remaining five questions,
d. (6 pts) There are five sections of the introductory psychology
course and each section has 100
students. Suppose that none of the students in section 2 studied for the
first quiz. If each of the
students in section 2 answered each of the 10 questions by randomly
selecting one of the four
choices,
What is the expected number of passing quiz scores in section 2‘s
student distribution of
quiz scores? __________
What is the expected value of section 2‘s quiz scores (in
percentages)? __________
What is the standard deviation of the quiz scores (in percentages)?
______________
------------------------------------------------------------------------------------
In order to implement a DBMS
In order to implement a DBMS, there must exist a set of rules which
state how the
database system will behave. For instance, somewhere in the DBMS
must be a set of
statements which indicate than when someone inserts data into a row
of a relation, it
has the effect which the user expects. One way to specify this is to use
words to write
an `essay' as to how the DBMS will operate, but words tend to be
imprecise and open to
interpretation. Instead, relational databases are more usually defined
using Relational

Algebra.
Relational Algebra is :
the formal description of how a relational database operates
an interface to the data stored in the database itself
the mathematics which underpin SQL operations
Operators in relational algebra are not necessarily the same as SQL
operators, even if
they have the same name. For example, the SELECT statement exists
in SQL, and also
exists in relational algebra. These two uses of SELECT are not the
same. The DBMS
must take whatever SQL statements the user types in and translate
them into relational
algebra operations before applying them to the database. Terminology
Relation - a set of tuples.
Tuple - a collection of attributes which describe some real world
entity.
Attribute - a real world role played by a named domain.
Domain - a set of atomic values.
Set - a mathematical definition for a collection of objects which
contains no
duplicates. Operators - Write
INSERT - provides a list of attribute values for a new tuple in a
relation. This
operator is the same as SQL.
DELETE - provides a condition on the attributes of a relation to
determine which
tuple(s) to remove from the relation. This operator is the same as
SQL.
MODIFY - changes the values of one or more attributes in one or
more tuples of
a relation, as identified by a condition operating on the attributes of
the relation.
This is equivalent to SQL UPDATE. Operators - Retrieval
There are two groups of operations:
Mathematical set theory based relations:
UNION, INTERSECTION, DIFFERENCE, and CARTESIAN

PRODUCT.
Special database operations:
SELECT (not the same as SQL SELECT), PROJECT, and JOIN.
Relational SELECT
SELECT is used to obtain a subset of the tuples of a relation that
satisfy a select
condition.
For example, find all employees born after 1st Jan 1950:
SELECTdob '01/JAN/1950'(employee) Relational PROJECT
The PROJECT operation is used to select a subset of the attributes of
a relation by
specifying the names of the required attributes.
For example, to get a list of all employees surnames and employee
numbers:
PROJECTsurname,empno(employee) SELECT and PROJECT
SELECT and PROJECT can be combined together. For example, to
get a list of
employee numbers for employees in department number 1: Figure :
Mapping select and project Set Operations - semantics
Consider two relations R and S. UNION of R and S
the union of two relations is a relation that includes all the tuples that
are either in
R or in S or in both R and S. Duplicate tuples are eliminated.
INTERSECTION of R and S
the intersection of R and S is a relation that includes all tuples that are
both in R
and S.
DIFFERENCE of R and S
the difference of R and S is the relation that contains all the tuples that
are in R
but that are not in S. SET Operations - requirements
For set operations to function correctly the relations R and S must be
union compatible.
Two relations are union compatible if
they have the same number of attributes
the domain of each attribute in column order is the same in both R
and S. UNION Example Figure : UNION INTERSECTION Example

Figure : Intersection DIFFERENCE Example Figure : DIFFERENCE
CARTESIAN PRODUCT
The Cartesian Product is also an operator which works on two sets. It
is sometimes
called the CROSS PRODUCT or CROSS JOIN.
It combines the tuples of one relation with all the tuples of the other
relation. CARTESIAN PRODUCT example Figure : CARTESIAN
PRODUCT JOIN Operator
JOIN is used to combine related tuples from two relations:
In its simplest form the JOIN operator is just the cross product of the
two
relations.
As the join becomes more complex, tuples are removed within the
cross product
to make the result of the join more meaningful.
JOIN allows you to evaluate a join condition between the attributes of
the
relations on which the join is undertaken.
The notation used is
R JOINjoin condition S JOIN Example Figure : JOIN Natural Join
Invariably the JOIN involves an equality test, and thus is often
described as an equi-join.
Such joins result in two attributes in the resulting relation having
exactly the same value.
A `natural join' will remove the duplicate attribute(s).
In most systems a natural join will require that the attributes have the
same name
to identify the attribute(s) to be used in the join. This may require a
renaming
mechanism.
If you do use natural joins make sure that the relations do not have
two attributes
with the same name by accident. OUTER JOINs
Notice that much of the data is lost when applying a join to two
relations. In some cases
this lost data might hold useful information. An outer join retains the
information that

would have been lost from the tables, replacing missing data with
nulls.
There are three forms of the outer join, depending on which data is to
be kept.
LEFT OUTER JOIN - keep data from the left-hand table
RIGHT OUTER JOIN - keep data from the right-hand table
FULL OUTER JOIN - keep data from both tables OUTER JOIN
example 1 Figure : OUTER JOIN (left/right) OUTER JOIN example
2 Figure : OUTER JOIN (full)
------------------------------------------------------------------------------------
MATH 6A Evaluate the line integral
2.Evaluate the line integral ∫C7xy4ds, where C is the right half of the
circle x2+y2=36
3.Evaluate the line integral ∫CF⋅dr,
where F(x,y,z)=−sinxi−2cosyj+4xzk and C is given by the vector
function r(t)=t5i−t4j+t3k , 0≤t≤1.
4.Sketch the vector field F⃗ (x,y)=xj⃗ , the line segment
from (1,4) to (7,4), and the line segment from (4,5) to(4,7).
(a) Calculate the line integral of the vector field F⃗ along the line
segment from (1,4) to (7,4).
(b) Calculate the line integral of the vector field F⃗ along the line
segment from (4,5) to (4,7).
5.Suppose F⃗ (x,y)=−ysin(x)i⃗ +cos(x)j⃗ .
(a) Find a vector parametric equation for the parabola y=x2 from the
origin to the point (4,16) using t as a parameter.
r⃗ (t)=
(b) Find the line integral of F⃗ along the parabola y=x2 from the
origin to (4,16).

6.Suppose F⃗ (x,y)=−yi⃗ +xj⃗ and C is the line segment from
point P=(5,0) to Q=(0,3).
(a) Find a vector parametric equation r⃗ (t) for the line segment C so
that points P and Q correspond to t=0 and t=1, respectively.
r⃗ (t)=
(b) Using the parametrization in part (a), the line integral
of F⃗ along C is
∫CF⃗ ⋅dr⃗ =∫baF⃗ (r⃗ (t))⋅r⃗ ′(t)dt=∫badt
with limits of integration a=and b=
(c) Evaluate the line integral in part (b).
(d) What is the line integral of F⃗ around the clockwise-
oriented triangle with corners at the origin, P, and Q? Hint: Sketch the
vector field and the triangle.
7.
Use a CAS to calculate ∫⟨ex−y,ex+y⟩⋅ds to four decimal places,
where  is the curve y=sinx for 0≤x≤π9, oriented from left to right.
Answer:
8.
Evaluate the line integral∫Cydx+xdywhereCis the parameterized
pathx=t3,y=t2,2≤t≤5.
∫Cydx+xdy=
9.LetCbe the straight path from(0,0)to(5,5)and
letF⃗ =(y−x−4)i⃗ +(sin(y−x)−4)j⃗ .
(a) At each point of C, what angle does F⃗ make with a tangent vector
to C?
angle =
(Give your answer in radians.)
(b) Find the magnitude ∥F⃗ ∥ at each point of C.
∥F⃗ ∥=

Evaluate ∫CF⃗ ⋅dr⃗ .
∫CF⃗ ⋅dr⃗ =
10. Let C be the curve which is the union of two line segments, the
first going from (0, 0) to (-3, 1) and the second going from (-3, 1) to (-
6, 0).
Compute the line integral ∫C−3dy−1dx.
15.Suppose ∇f(x,y)=5ysin(xy)i⃗ +5xsin(xy)j⃗ , F⃗ =∇f(x,y), and C is
the segment of the parabola y=4x2from the point (3,36) to (4,64).
Then
∫CF⃗ ⋅dr⃗
16.Suppose F⃗ (x,y)=(x+4)i⃗ +(3y+2)j⃗ . Use the fundamental theorem
of line integrals to calculate the following.
(a) The line integral of F⃗ along the line segment C from the
point P=(1,0) to the point Q=(4,3).
∫CF⃗ ⋅dr⃗ =
(b) The line integral of F⃗ along the triangle C from the origin to the
point P=(1,0) to the point Q=(4,3) and back to the origin.
∫CF⃗ ⋅dr⃗ =
20.For the vector
field G⃗ =(yexy+5cos(5x+y))i⃗ +(xexy+cos(5x+y))j⃗ , find the line
integral of G⃗ along the curve C from the origin along the x-axis to
the point (5,0) and then counterclockwise around the circumference of
the circle x2+y2=25 to the point (5/2√,5/2√).∫CG⃗ ⋅dr⃗
21.
------------------------------------------------------------------------------------
MATH 201 In a public opinion poll

3) In a public opinion poll, approximately 85% of Americans felt that
having a police officer on patrol in the neighborhood made them safer
than having no police officer on patrol. The margin of error reported
was 3%. Construct an interval estimate using these figures.
4) For each of the following confidence levels, look up the critical z
values for a two-tailed test.
4a) 90% (Hint: 5% in each tail):
Work:
4b) 95% (Hint: 2.5% in each tail):
Work:
5)Remembering that a meta-analysis calculates one mean effect size
using the effect sizes of several studies, assume you are conducting a
meta-analysis over a set of five studies. The effect sizes for each study
are: d= .65, d = .19, d = .42, d = .08, d = .70
5a) Calculate the mean effect size of these studies.
5b) Use Cohen's conventions to describe the mean effect size you
calculated in part (a).
------------------------------------------------------------------------------------
MATH 333 Matrix Algebra and Complex Variables
Homework 5
Note: show full steps to get full credit. 1. Consider function f (z) = z 3
.
(a) Use the definition of derivatives to calculate the derivative of f ,
the result should be a function
of z.

(b) Use Cauchy-Riemann equation to show that f is an entire function.
(c) Find the derivative of f (z) in terms of u and v.
2. At what points are the following functions not analytic
(a) z
z−3i . (b) z 2 −2iz
.
z 2 +4 3. Compute f 0 if f = 4z 3 −5z+1
2z−1 . 4. Show that the following functions are not analytic at any
point.
(a) f (z) = y + ix
(b) f (z) = z¯2
(c) f (z) = 2x2 + y + i(y 2 − x)
5. Show that (ez )0 = ez .
6. Express f (z) = e2¯z in u + iv form.
7. Solve ez−1 = −ie2 .
8. Express sin(−2i) in a + ib form.
9. Find all values of z such that cos z = −3i. 1
------------------------------------------------------------------------------------
PRINTABLE VERSION Quiz 8 Question 1 1 x + + 3
Quiz 8
Question 1 1
πx + π + 3 .
(5
) Find the period for the following function: f (x) = 8 cos a) 2π
5 b) π
5 c) 10 d) 5 e) 10 π f) None of the above. Question 2
Find the phase shift for the following function: f (x) = 3 sin a) π units
to the left b) 3 units to the right c) 3 units to the left d) 2 units to the
left e) π units to the right f) None of the above. Question 3 1
πx + π − 2 .
(3

) Give an equation of the form f (x) = A cos(Bx − C) + D which could
be used to represent the given
graph. (Note: C or D may be zero.) a) f (x) = 3 cos(2 x − π) b) f (x) =
3 cos(2 x − π) − 1 c) f (x) = 6 cos(2 x − π) + 1 d) f (x) = 3 cos(2 x − π)
+ 1 e) f (x) = 6 cos(2 x − π) − 1 f) None of the above. Question 4
Give an equation of the form f (x) = A sin(Bx − C) + D which could
be used to represent the given
graph. (Note: C or D may be zero.) a) f (x) = 3 sin(4 x − π
2) b) f (x) = 3 sin(4 x − π
−1
2) c) f (x) = 6 sin(4 x − π
+1
2) d) f (x) = 6 sin(4 x − π
−1
2) e) f (x) = 3 sin(4 x − π
+1
2) f) None of the above. Question 5
List all x -intercepts for y = −3 cos(4x + a) − π 5π
,
12 12 π
π π
on
the
interval
−
[ 6 , 2 ].
3) b) π 5π
,
12 24 c) π π
,
24 4 d) 0, e) π 7π
,
24 24 f) None of the above. 7π
24 Question 6
Write a sine function with a positive vertical displacement given the
following information: the amplitude
is 6 , the horizontal shift is 12 to the left, y − intercept is (0, 3), and

the period is 8 . 1
x + 3π − 3
(4
) a) f (x) = 6 sin b) f (x) = 6 sin( π
x − 3 π) − 3
4 c) f (x) = 6 sin( π
x + 3 π) + 3
8 d) f (x) = 6 sin( π
x + 3 π) + 3
4 e) f (x) = 6 sin f) None of the above. 1
x − 3π + 3
(4
) Question 7
The current I , in amperes, flowing through an AC (alternating
current) circuit at time t is I(t) = 200 sin(35 πt + a) Period: π
, where t ≥ 0 . Find the period and the horizontal shift.
6) 1
1
, shift
to the left.
35
105 b) Period: 1
1
, shift
to the right.
35
105 c) Period: 2
1
, shift
to the left.
35
420 d) Period: 2
1
, shift
to the right.
35
210 e) Period: 2

------------------------------------------------------------------------------------
Prove a fundamental result about polynomials
We will now prove a fundamental result about polynomials: every
non-zero polynomial of degreen (over a field F) has at most n roots. If
you don‘t know what a field is, you can assume in thefollowing that F
= R (the real numbers).
(a) Show that for any α ∈ F, there exists some polynomial Q(x) of
degree n−1 and some b ∈ Fsuch that P(x) = (x−α)Q(x) +b.
(b) Show that if α is a root of P(x), then P(x) = (x−α)Q(x).
(c) Prove that any polynomial of degree 1 has at most one root. This is
your base case.(d) Now prove the inductive step: if every polynomial
of degree n−1 has at most n−1 roots, thenany polynomial of degree n
has at most n roots
------------------------------------------------------------------------------------
QNT 295 The H2 Hummer limousine has eight tires on it
The H2 Hummer limousine has eight tires on it. A fleet of 1230 H2
limos was fit with a batch of tires that mistakenly passed quality
testing. The following table lists the frequency distribution of the
number of defective tires on the 1230 H2 limos.
Number of defective tires
0 1 2 3 4 5 6 7 8

Number of H2 limos 50 228 333 328 195
76 16 3 1
Construct a probability distribution table for the numbers of defective
tires on these limos.
Round your answers to three decimal places.
x P(x)
0 0
1 228
2 666
3 904
4 780
5 380
6 96
7 21
8 8
Calculate the mean and standard deviation for the probability
distribution you developed for the number of defective tires on all
1230 H2 Hummer limousines. Round your answers to three decimal
places. There is an average of 342.55 defective tires per limo, with a
standard deviation of 18.50 tires.
The H2 Hummer limousine has eight tires on it. A fleet of 1230 H2
limos was fit with a batch of tires that mistakenly passed quality
testing. The following table lists the frequency distribution of the
number of defective tires on the 1230 H2 limos.
Number of defective tires

0 1 2 3 4 5 6 7 8
Number of H2 limos 50 228 333 328 195
76 16 3 1
Construct a probability distribution table for the numbers of defective
tires on these limos. Round your answers to three decimal places.
x P(x)
0 0
1 228
2 666
3 904
4 780
5 380
6 96
7 21
8 8
Calculate the mean and standard deviation for the probability
distribution you developed for the number of defective tires on all
1230 H2 Hummer limousines. Round your answers to three decimal
places. There is an average of 342.55 defective tires per limo, with a
standard deviation of 18.50 tires.
------------------------------------------------------------------------------------
Question numbers refer to the exercises at the end of each
section of the textbook

Question numbers refer to the exercises at the end of each section of
the textbook (1St edition numbers are given if they are different). You
must show your work to get full marks. For
some questions, I have given hints, clarifications, or extra instructions.
Be sure to follow these! Note: Only a selection of exercises may be
graded. 1) A reflection in R2 across a line I. through the origin is a
linear transformation that
takes a vector 56 to its ―mirror image‖ on the opposite side of L.
Suppose that M is the standard matrix for reflection across L. L
a. What happens to any 56 if you apply the reflection ‗- g}
transformation twice?
b. Explain why this tells you that M 2 = I, must be true. a 3 c. Show
that it follows that M '1 = M . d. Bonus: It is a remarkable fact that
doing a reflection across one line followed by a reflection across
another line always ends up being a rotation. This can be proven with
matrix multiplication, as follows. The matrix for a reflection across
the line that is inclined at angle a is given _ cos(2a) sin(2a)
by M― ' sin(2a) -cos(2a) ‘ and the matrix for a rotation by angle 6' is
given by R9 = sin 6 cos 6 cos!) — sing ]_
Use matrix multiplication and trigonometric identities to prove that
the composition of a reflection across the line with angle a with
another reflection across the line with angle 5 is a rotation. What is
the angle of this rotation? 2) Do textbook question 3.3 #54. a
3) Let S be the subset of R4 consisting of all vectors that have the
form 3; 3g , b where a and b are any scalars.
a. Find four different vectors that are in S. You should show how you
know they are in S. Then write down one vector in R4 that is not in S,
with an explanation
of how you know that. b. Prove that S is a subspace of R4 by proving
that it satisfies all three
conditions of a subspace. Then find a basis for the subspace (with
some explanation of how you know it is a basis) and the dimension of
the subspace
------------------------------------------------------------------------------------

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