Topology
2nd Edition
ISBN: 9780134689517
Author: Munkres, James R.
Publisher: Pearson,
expand_more
expand_more
format_list_bulleted
Videos
Textbook Question
Chapter 1.10, Problem 5E
Show the well-ordering theorem implies the choice axiom.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
Determine whether it's true or false and the reasoning is needed
1. (20 pts) Determine whether the following statements are true (T) or false (F)? (A
reasoning is required.)
(1) Let V be the set of all ordered pairs of real numbers. Consider the following
addition and scalar multiplication operations on u =
u= (u1, u2) and v =
(v1, v2): u + v = (U₁ + V₁, U₂ + v₂), ku = (ku₁, u₂). Is V a vector space
under the above operations?
U2
(2) The set Mmxn of all m×n matrices with the usual operations of addition
and scalar multiplication is a vector space.
α
(3) The dimension of the vector space of all matrices A = [a b] in R2×2 with
a+d=0 is 4.
(4) The coordinate vector of p(x) = 2-x+x² in P3 relative to the basis S =
{1, 1+x, x + x2} is [4 -2 1].
(5) If a 6×4 matrix A has a rank 3, then the dimension of N(A) is 3.
5. (20%) The linear transformation L: P3 → P2 defined by L(f(x)) = f'(x)+
f(0).
(a) Find the representing matrix A of L with respect to the ordered basis
{x2, x, 1} for P3, and the ordered basis {2,1 - x} for P2.
(b) Find the coordinates of the f(x) = 2x² +2 in P3 with respect to the
ordered basis {x2,-x, 1}, and find the coordinates of L(f(x)) with respect
to the ordered basis {2,1-x}
Chapter 1 Solutions
Topology
Ch. 1.1 - Check the distributive laws for and and De Morgans...Ch. 1.1 - Determine which of the following statements are...Ch. 1.1 - Determine which of the following statements are...Ch. 1.1 - Determine which of the following statements are...Ch. 1.1 - Determine which of the following statements are...Ch. 1.1 - Determine which of the following statements are...Ch. 1.1 - Determine which of the following statements are...Ch. 1.1 - Determine which of the following statements are...Ch. 1.1 - Determine which of the following statements are...Ch. 1.1 - Determine which of the following statements are...
Ch. 1.1 - Determine which of the following statements are...Ch. 1.1 - Prob. 2.11ECh. 1.1 - Determine which of the following statements are...Ch. 1.1 - Determine which of the following statements are...Ch. 1.1 - Determine which of the following statements are...Ch. 1.1 - Determine which of the following statements are...Ch. 1.1 - Determine which of the following statements are...Ch. 1.1 - Determine which of the following statements are...Ch. 1.1 - Write the contrapositive and converse of the...Ch. 1.1 - Do the same for the statement If x0, then x2x0.Ch. 1.1 - Let A and B be sets of real numbers. Write the...Ch. 1.1 - Let A and B be sets of real numbers. Write the...Ch. 1.1 - Let A and B be sets of real numbers. Write the...Ch. 1.1 - Let A and B be sets of real numbers. Write the...Ch. 1.1 - Let A be a nonempty collection of sets. Determine...Ch. 1.1 - Write the contrapositive of each of the statements...Ch. 1.1 - Write the contrapositive of each of the statements...Ch. 1.1 - Write the contrapositive of each of the statements...Ch. 1.1 - Write the contrapositive of each of the statements...Ch. 1.1 - Prob. 7ECh. 1.1 - Prob. 8ECh. 1.1 - Formulate and prove DeMorgans laws for arbitrary...Ch. 1.1 - Let denote the set of real numbers. For each of...Ch. 1.1 - Let denote the set of real numbers. For each of...Ch. 1.1 - Let denote the set of real numbers. For each of...Ch. 1.1 - Let denote the set of real numbers. For each of...Ch. 1.1 - Let denote the set of real numbers. For each of...Ch. 1.2 - Let f:AB. Let A0AandB0B. Show that A0f1(f(A0)) and...Ch. 1.2 - Prob. 1.2ECh. 1.2 - Let f:AB and let AiAandBiBfori=0andi=1. Show that...Ch. 1.2 - Let f:AB and let AiAandBiBfori=0andi=1. Show that...Ch. 1.2 - Let f:AB and let AiAandBiBfori=0andi=1. Show that...Ch. 1.2 - Let f:AB and let AiAandBiBfori=0andi=1. Show that...Ch. 1.2 - Prob. 2.5ECh. 1.2 - Let f:AB and let AiAandBiBfori=0andi=1. Show that...Ch. 1.2 - Let f:AB and let AiAandBiBfori=0andi=1. Show that...Ch. 1.2 - Let f:AB and let AiAandBiBfori=0andi=1. Show that...Ch. 1.2 - Show that b, c, f, and g of Exercise 2 hold for...Ch. 1.2 - Show that b, c, f, and g of Exercise 2 hold for...Ch. 1.2 - Show that b, c, f, and g of Exercise 2 hold for...Ch. 1.2 - Show that b, c, f, and g of Exercise 2 hold for...Ch. 1.2 - Let f:AB and g:BC. If C0C, show that...Ch. 1.2 - Let f:AB and g:BC. If f and g are injective, show...Ch. 1.2 - Let f:AB and g:BC. If gf is injective, what can...Ch. 1.2 - Let f:AB and g:BC. If f and g are surjective, show...Ch. 1.2 - Let f:AB and g:BC. If gf is surjective, what can...Ch. 1.2 - Let f:AB and g:BC. Summarize your answers to b-e...Ch. 1.2 - In general, let us denote the identity function...Ch. 1.2 - In general, let us denote the identity function...Ch. 1.2 - In general, let us denote the identity function...Ch. 1.2 - In general, let us denote the identity function...Ch. 1.2 - In general, let us denote the identity function...Ch. 1.2 - Let f: be the function f(x)=x3x. By restricting...Ch. 1.3 - Define two points (x0,y0) and (x1,y1) of the plane...Ch. 1.3 - Let C be a relation on a set A. If A0A, define the...Ch. 1.3 - Here is a proof that every relation C that is both...Ch. 1.3 - Let f:AB be a surjective function. Let us define a...Ch. 1.3 - Let f:AB be a surjective function. Let us define a...Ch. 1.3 - Let S and S be the following subsets of the plane:...Ch. 1.3 - Let S and S be the following subsets of the plane:...Ch. 1.3 - Let S and S be the following subsets of the plane:...Ch. 1.3 - Define a relation on the plane by setting...Ch. 1.3 - Show that the restriction of an order relation is...Ch. 1.3 - Check that the relation defined in Example 7 is an...Ch. 1.3 - Check that the dictionary order is an order...Ch. 1.3 - a Show that the map f:(1,1) of Example 9 is order...Ch. 1.3 - Prob. 10.2ECh. 1.3 - Prob. 11ECh. 1.3 - Prob. 12ECh. 1.3 - Prove the following: Theorem. If an ordered set A...Ch. 1.3 - If C is a relation on a set A, define a new...Ch. 1.3 - Assume that the real line has the least upper...Ch. 1.4 - Prove the following laws of algebra for , using...Ch. 1.4 - Prove the following laws of algebra for , using...Ch. 1.4 - Prob. 1.3ECh. 1.4 - Prob. 1.4ECh. 1.4 - Prob. 1.5ECh. 1.4 - Prob. 1.6ECh. 1.4 - Prove the following laws of algebra for , using...Ch. 1.4 - Prove the following laws of algebra for , using...Ch. 1.4 - Prob. 1.9ECh. 1.4 - Prob. 1.10ECh. 1.4 - Prob. 1.11ECh. 1.4 - Prob. 1.12ECh. 1.4 - Prob. 1.13ECh. 1.4 - Prob. 1.14ECh. 1.4 - Prob. 1.15ECh. 1.4 - Prob. 1.16ECh. 1.4 - Prove the following laws of algebra for , using...Ch. 1.4 - Prob. 1.18ECh. 1.4 - Prob. 1.19ECh. 1.4 - Prob. 1.20ECh. 1.4 - Prob. 2.1ECh. 1.4 - Prob. 2.2ECh. 1.4 - Prob. 2.3ECh. 1.4 - Prob. 2.4ECh. 1.4 - Prob. 2.5ECh. 1.4 - Prob. 2.6ECh. 1.4 - Prob. 2.7ECh. 1.4 - Prob. 2.8ECh. 1.4 - Prob. 2.9ECh. 1.4 - Prob. 2.10ECh. 1.4 - Prob. 2.11ECh. 1.4 - Prob. 3ECh. 1.4 - Prob. 4.1ECh. 1.4 - Prob. 4.2ECh. 1.4 - Prove the following properties of and+: a...Ch. 1.4 - Prob. 6ECh. 1.4 - Prob. 7ECh. 1.4 - Prob. 8.1ECh. 1.4 - Prob. 8.2ECh. 1.4 - Prob. 8.3ECh. 1.4 - a Show that every nonempty subset of that is...Ch. 1.4 - Prob. 10.1ECh. 1.4 - Prob. 10.2ECh. 1.4 - Prob. 10.3ECh. 1.4 - Prob. 10.4ECh. 1.4 - Prob. 11.1ECh. 1.4 - Prob. 11.2ECh. 1.4 - Prob. 11.3ECh. 1.4 - Prob. 11.4ECh. 1.5 - Show there is a bijective correspondence of AB...Ch. 1.5 - a Show that if n1 there is bijective...Ch. 1.5 - b Given the indexed family {A1,A2,}, let...Ch. 1.5 - Let A=A1A2 and B=B1B2. a Show that if BiAi for all...Ch. 1.5 - Let A=A1A2 and B=B1B2. b Show the converse of a...Ch. 1.5 - Let A=A1A2 and B=B1B2. c Show that if A is...Ch. 1.5 - Prob. 3.4ECh. 1.5 - Let m,n+. Let X. a If mn, find an injective map...Ch. 1.5 - Let m,n+. Let X. b Find a bijective map...Ch. 1.5 - Let m,n+. Let X. c Find an injective map h:XnX.Ch. 1.5 - Let m,n+. Let X. d Find a bijective map k:XnXX.Ch. 1.5 - Prob. 4.5ECh. 1.5 - Prob. 4.6ECh. 1.5 - Which of the following subsets of can be...Ch. 1.6 - a Make a list of all the injective maps...Ch. 1.6 - Prob. 2ECh. 1.6 - Prob. 3ECh. 1.6 - Prob. 4.1ECh. 1.6 - Prob. 4.2ECh. 1.6 - If AB is finite, does it follow that A and B are...Ch. 1.6 - a Let A={1,,n}. Show there is a bijection of P(A)...Ch. 1.6 - b Show that if A is finite, then P(A) is finite.Ch. 1.6 - Prob. 7ECh. 1.7 - Show that is countably infinite.Ch. 1.7 - Show that the maps f and g of Examples 1 and 2 are...Ch. 1.7 - Prob. 3ECh. 1.7 - a A real number x is said to be algebraic over the...Ch. 1.7 - Determine, for each of the following sets, whether...Ch. 1.7 - Determine, for each of the following sets, whether...Ch. 1.7 - Determine, for each of the following sets, whether...Ch. 1.7 - Determine, for each of the following sets, whether...Ch. 1.7 - Determine, for each of the following sets, whether...Ch. 1.7 - Determine, for each of the following sets, whether...Ch. 1.7 - Determine, for each of the following sets, whether...Ch. 1.7 - Determine, for each of the following sets, whether...Ch. 1.7 - Prob. 5.9ECh. 1.7 - Prob. 5.10ECh. 1.7 - We say that two sets A and B have the same...Ch. 1.7 - We say that two sets A and B have the same...Ch. 1.7 - Show that the sets D and E of Exercise 5 have the...Ch. 1.7 - Let X denote the two-element set {0,1}; let B be...Ch. 1.7 - a The formula...Ch. 1.8 - Prob. 1ECh. 1.8 - Prob. 2ECh. 1.8 - Prob. 3ECh. 1.8 - Prob. 4ECh. 1.8 - Prob. 5ECh. 1.8 - Prob. 6ECh. 1.8 - Prob. 7ECh. 1.8 - Prob. 8ECh. 1.9 - Define an injective map f:+X, where X is the...Ch. 1.9 - Prob. 2ECh. 1.9 - Prob. 3ECh. 1.9 - There was a theorem in 7 whose proof involved an...Ch. 1.9 - a Use the choice axiom to show that if f:AB is...Ch. 1.9 - Let A and B be two nonempty sets. If there is an...Ch. 1.9 - Prob. 8ECh. 1.10 - Prob. 1ECh. 1.10 - Both {1,2}+ and +{1,2} are well-ordered in the...Ch. 1.10 - a Let denote the set of negative integers in the...Ch. 1.10 - Show the well-ordering theorem implies the choice...Ch. 1.10 - Prob. 6ECh. 1.10 - a Let A1 and A2 be disjoint sets, well-ordered by...Ch. 1.10 - Let A and B be two sets. Using the well-ordering...
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Similar questions
- For the spinner below, assume that the pointer can never lie on a borderline. Find the following probabilities. (enter the probabilities as fractions)arrow_forwardQuestions 1. Identify and describe potential bias in the study. 2. Identify and describe the way in which the selected participants may or may not represent the population as a whole. 3. Identify and describe the possible problems with the end results since the majority will be from females rather than an even split. 4. Identify and describe the possible problems with identifying females as possibly more vulnerable based on the data collected. 5. Identify a possible null hypothesis and problems in how the study might address this null hypothesis. 6. Identify one possible method of improving the study design and describe how it would improve the validity of the conclusions. 7. Identify a second possible method of improving the study design and describe how it would improve the validity of the conclusions.arrow_forwardThe Course Name Real Analysis please Solve questions by Real Analysisarrow_forward
- part 3 of the question is: A power outage occurs 6 min after the ride started. Passengers must wait for their cage to be manually cranked into the lowest position in order to exit the ride. Sine function model: where h is the height of the last passenger above the ground measured in feet and t is the time of operation of the ride in minutes. What is the height of the last passenger at the moment of the power outage? Verify your answer by evaluating the sine function model. Will the last passenger to board the ride need to wait in order to exit the ride? Explain.arrow_forward2. The duration of the ride is 15 min. (a) How many times does the last passenger who boarded the ride make a complete loop on the Ferris wheel? (b) What is the position of that passenger when the ride ends?arrow_forward3. A scientist recorded the movement of a pendulum for 10 s. The scientist began recording when the pendulum was at its resting position. The pendulum then moved right (positive displacement) and left (negative displacement) several times. The pendulum took 4 s to swing to the right and the left and then return to its resting position. The pendulum's furthest distance to either side was 6 in. Graph the function that represents the pendulum's displacement as a function of time. Answer: f(t) (a) Write an equation to represent the displacement of the pendulum as a function of time. (b) Graph the function. 10 9 8 7 6 5 4 3 2 1 0 t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 -1 -5. -6 -7 -8 -9 -10-arrow_forward
- A power outage occurs 6 min after the ride started. Passengers must wait for their cage to be manually cranked into the lowest position in order to exit the ride. Sine function model: h = −82.5 cos (3πt) + 97.5 where h is the height of the last passenger above the ground measured in feet and t is the time of operation of the ride in minutes. (a) What is the height of the last passenger at the moment of the power outage? Verify your answer by evaluating the sine function model. (b) Will the last passenger to board the ride need to wait in order to exit the ride? Explain.arrow_forwardThe Colossus Ferris wheel debuted at the 1984 New Orleans World's Fair. The ride is 180 ft tall, and passengers board the ride at an initial height of 15 ft above the ground. The height above ground, h, of a passenger on the ride is a periodic function of time, t. The graph displays the height above ground of the last passenger to board over the course of the 15 min ride. Height of Passenger in Ferris Wheel 180 160 140- €120 Height, h (ft) 100 80 60 40 20 0 ך 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Time of operation, t (min) Sine function model: h = −82.5 cos (3πt) + 97.5 where h is the height of the passenger above the ground measured in feet and t is the time of operation of the ride in minutes. What is the period of the sine function model? Interpret the period you found in the context of the operation of the Ferris wheel. Answer:arrow_forward1. Graph the function f(x)=sin(x) −2¸ Answer: y -2π 一元 1 −1 -2 -3 -4+ 元 2πarrow_forward
- 3. Graph the function f(x) = −(x-2)²+4 Answer: f(x) 6 5 4 3 2+ 1 -6-5 -4-3-2-1 × 1 2 3 4 5 6 -1 -2+ ရာ -3+ -4+ -5 -6arrow_forward2. Graph the function f(x) = cos(2x)+1 Answer: -2π 一元 y 3 2- 1 -1 -2+ ရာ -3- Π 2πarrow_forward2. Graph the function f(x) = |x+1+2 Answer: -6-5-4-3-2-1 f(x) 6 5 4 3 2 1 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning
Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning
Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw Hill
Elementary Geometry For College Students, 7eGeometryISBN:9781337614085Author:Alexander, Daniel C.; Koeberlein, Geralyn M.Publisher:Cengage,
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:9781305658004
Author:Ron Larson
Publisher:Cengage Learning
Trigonometry (MindTap Course List)
Trigonometry
ISBN:9781337278461
Author:Ron Larson
Publisher:Cengage Learning
Glencoe Algebra 1, Student Edition, 9780079039897...
Algebra
ISBN:9780079039897
Author:Carter
Publisher:McGraw Hill
Elementary Geometry For College Students, 7e
Geometry
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Cengage,
What is a Relation? | Don't Memorise; Author: Don't Memorise;https://www.youtube.com/watch?v=hV1_wvsdJCE;License: Standard YouTube License, CC-BY
RELATIONS-DOMAIN, RANGE AND CO-DOMAIN (RELATIONS AND FUNCTIONS CBSE/ ISC MATHS); Author: Neha Agrawal Mathematically Inclined;https://www.youtube.com/watch?v=u4IQh46VoU4;License: Standard YouTube License, CC-BY