Connor Galvin
NICO pg 176
41.
a) cfw_x: x is a natural number divisible by 2 or 3
Bijection: a bijection exists if
f(a)=a when a is even
f(a)=a/3 when a is odd
b) cfw_x: x is real and x - LxJ = 0 or 0.5
Bijection when:
f(a)= 2a
C) The set of words over cf
Connor Galvin
Intro to Proofs
Nicodemi
Verbal Proof
Demorgans Laws
~(p and q) <=> ~p or ~q
Assume that ~(P and Q) is true, so (P and Q) is false. Since (P and Q) is false, either P is false or Q
is false (or both). In the case that either p is false or q
Connor Galvin
Functions and Sets Worksheet
4e.
Proposition: F[V] = (F[V])
Example: F(x) = x2, A= cfw_1,2,3,4,5,6, V= cfw_1, V = cfw_ 2,3,4,5,6
F[V] = cfw_4,9,16,25,36 = (F[V])=cfw_4,9,16,25,36
Proof: Let z F[V]. So, we can find an x V such that f(x) = z.
Connor Galvin
12/4/2016
Cardinalities
6.
Proposition: Suppose that A and B are disjoint sets such that |A| = 0 and |B| = n. Prove that |A B|
= 0.
Proof: Let |A| = 0, and |B| = n, such that both sets are disjoint. So, |A B| = |A|+ |B| - |A B|. If
this is t
Connor Galvin
Professor Nicodemi
8/30/2016
Assignment #1
1,2,3,4,6b
1.
a)
b)
c)
d)
e)
f)
g)
Prop
Prop
Not prop
Not prop
Prop
Prop
Not prop
a)
b)
c)
d)
e)
f)
Mary is not tall
The Cat is black
Some dogs bark
All birds have feathers
Mary is not tall and Burt
Connor Galvin
Roomy sets hw
Nicodemi
3. Proposition: If A and B are roomy sets, then A B is roomy.
c
Proof: Let A and B be sets, such that both are roomy sets. If A is a roomy set, then for every x A,
there is a positive distance rx > 0 such that the open
Connor Galvin
NICO HW pg 217
Question 8:
1) Some related pairs include:
(3,9) R (1,3), (4,8) R (1,2), (1,2) R (4,8), (1,1) R (1,1)
2) Prove that this is an equivalence Relation:
Proof:
First we must show that the relation is reflexive. Let xZ and yN. For
Connor Galvin
12/6/2016
Cardinalities Part 2
Problem 7:
Proposition: Suppose that A and B are disjoint sets, both of cardinality 0. Prove |A B| = 0.
Play: In order to get |A B| = 0, we must find a way to create a bijection between A or B to AB, this
way t
Connor Galvin
11/29/2016
Congruence Proof
If x0 is a solution to ax b mod n and y x0 mod n, then y is also a solution to ax b mod n. Is the
converse true?
Proof: Let x0 be a solution to ax b mod n. Also, let y x0 mod n. If y x0 mod n, then y E[x0]n. If y
simplest form, the fraction is equal to (1) a 1 b (2) a
b (3) 2(a 1 b) (4) b a 9. An angle of radians is
congruent to an angle of (1) 135 (2) 225 (3) 315
(4) 405 7p 4 a b 2 b a 1 a 2 1 b 1 16 1 2 logx 1 4 a 3
k51 (k 1 1)2 fA p 2 B !225 2!216 Cumulative R
be drawn and that triangle is a right triangle. Note:
If mA 5 150, sin B 5 1 and mB 5 90. There is no
triangle with an obtuse angle and a right angle. Only
one triangle can drawn. In ABC, a 5 16, b 5 8, and A
5 30. We can use the Law of Sines to find sin
substitutions, diagrams, graphs, charts, etc. For all
questions in this part, a correct numerical answer
with no work shown will receive only 1 credit. 15. If
logb x 5 logb 3 1 2 logb 4 , express x in simplest
form. 16. a. Sketch the graph of y 5 2 sin x
spelling errors and the frequency of that number of
errors, that is, the number of essays that contained
that number of misspellings. Find the mean number
of spelling errors for these essays. Solution To find
the total number of spelling errors, first mul
words. Let xi represent the number of misspelled
words in a report and fi represent the number of
reports that contain xi misspelled words. The mean
of this set of data is the total number of misspelled
words divided by the number of reports.To find the
t
know the length of the side adjacent to the angle of
elevation and we want to know the height of the
tree, the length of the side opposite the angle of
elevation. tan u 5 tan 57 5 h 5 12 tan 57 h 5 18.478
We know the measures of two angles and the
include
Hourly wages: $6.90, $7.10, $7.50, $7.50, $8.25,
$9.30, $9.50, $10.00 8. Tips: $1.00, $1.50, $2.25,
$3.00, $3.30, $3.50, $4.00, $4.75, $5.00, $5.00,
$5.00 In 914, find the median and the first and
third quartiles for each set of data values. 9. 2, 3, 5,
8
shape of the triangle are determined. We can use
the Law of Cosines or the Law of Sines to find the
measures of the remaining parts of the triangle. Law
of Cosines: a2 5 b2 1 c2 2 2bc cos A cos A 5 b2 5 a2
1 c2 2 2ac cos B cos B 5 c2 5 a2 1 b2 2 2ab cos C
on the ground 50 feet from the foot of a vertical
monument, the measure of the angle of elevation of
the top of the monument is 65 degrees. What is the
height of the monument to the nearest foot? 26. A
vertical telephone pole that is 15 feet high is brace
and a 5 25.8. Find c to the nearest tenth. 13. In PQR,
mP 5 125, mQ 5 14, and p 5 122. Find r to the
nearest integer. 14. In RST, mR 5 12, mS 5 75, and r 5
3.52. Find t to the nearest tenth. 15. In CDE, mD 5
125, mE 5 28, and d 5 12.5. Find c to the neare
effective. 3. The selection should be random or
determined in such a way as to eliminate any bias. If
a new medicine being tested is proposed for use by
people of all ages, of different ethnic backgrounds,
and for use by both men and women, then the
sampl
correspond with the stems of the stem-and-leaf
diagram. The frequencies can be determined by the
use of a tally to represent each price. c. The data set
is obtained from a random selection of stores from
all of the stores in the study and is therefore a
s
Ambiguous Case 573 P R 10 Q 110 12 5 10 sin R 5
12 sin 110 sin R 5 sin R 1.127631145 12 sin 1108
10 10 sin 1108 5 12 sin R r sin R p sin P P R 10 Q
110 h 12 70 14411C14.pgs 3/3/09 2:14 PM Page
573 Developing Skills In 314: a. Determine the
number of possi
park. Two of the angles at which the streets
intersect measure 85 degrees and 65 degrees. The
length of the longest side of the park is 275 feet.
Find the lengths of the other two sides of the park
to the nearest tenth. 21. On the playground, the 10foot l
integer, the area of the rhombus. 13. Use the Law of
Cosines to find two possible lengths for AB of ABC if
BC 5 7, AC 5 8, and mA 5 60. 14. Use the Law of
Sines to show that there are two possible triangles if
BC 5 7, AC 5 8, and mA 5 60. 15. A vertical p
value. 3. Draw a box with opposite sides through
the lower and upper quartiles and a vertical line
through the median. 4. Draw whiskers by drawing a
line to join the dot that represents the minimum
value to the dot that represents the lower quartile
and a
the angle measures and lengths of the sides of the
garden(s) if any. 17. Emily wants to draw a
parallelogram with the measure of one side 12
centimeters, the measure of one diagonal 10
centimeters and the measure of one angle 120
degrees. Is this possible
Find the mean, the median, and the mode of the
following set of grades: 92, 90, 90, 90, 88, 87, 85, 70
Solution Mean 5 5 (92 1 90 1 90 1 90 1 88 1 87 1 85
1 70) 5 5 86.5 Median 5 the average of the 4th and
5th grades 5 5 89 Mode 5 the grade that appears
m
the data when finding the first and third quartiles.
For example, the heights, in inches, of 19 children
are given below: 37, 39, 40, 42, 42, 43, 44, 44, 44,
45, 46, 47, 47, 48, 49, 49, 50, 52, 53 There are 19 5
2(9) 1 1 data values. Therefore, the second
sides. Given: Two sides and an angle opposite one
of them Use the Law of Sines to find the possible
measure(s) of another angle. Determine if there
are two, one, or no possible triangles. If there is a
triangle, use the sum of the angles of a triangle to
triangle are 22, 46, and 58. Find, to the nearest
degree, the measure of the largest angle of the
triangle. 5. Use the Law of Cosines to show that if
the measures of the sides of a triangle are 10, 24,
and 26, the triangle is a right triangle. A B C h a b