Homework Answers
Consider the function 
by 
, where 
is the ceiling function ( 
, 
, 
, 
,... )
That is 
if n is odd and 
if n is even. Therefore we can restate the function as
.
Clearly f is a well defined function (unique image for every
natural numbers) every element in C has unique pre-image. Hence f
is a bijection between 
and C
and thus C is countable.
Add Answer to:
9. Show that the set C N × a, b is countable by constructing a bijection...
Let S be a finite set with cardinality n>0. a. Prove, by constructing a bijection, that...
Let S be a finite set with cardinality n>0. a. Prove, by constructing a bijection, that the number of subsets of S of size k is equal to the number of subsets of size n- k. Be sure to prove that vour mapping is both injective and surjective. b. Prove, by constructing a bijection, that the number of odd-cardinality subsets of S is equal to the number of even-cardinality subsets of S. Be sure to prove that your mapping is...
3. (8 marks) Let be the set of integers that are not divisible by 3. Prove that is a countable set by finding a bijection between the set and the set of integers , which we know is countable from cl...
3. (8 marks) Let be the set of integers that are not divisible by 3. Prove that is a countable set by finding a bijection between the set and the set of integers , which we know is countable from class. (You need to prove that your function is a bijection.) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
2. Determine whether the given sets are countable or uncountable. Justify each answer with a bijection...
2. Determine whether the given sets are countable or uncountable. Justify each answer with a bijection (or table like we did with Q+) or using results from class/textbook. (a) {0, 1, 2} * N (b) A = {(x, y) : x2 + y2 = 1} (c) {0, 1} R Che set of all 2-element subsets of N (e) Real numbers with decimal representations consists of all 1s. (f) The set of all functions from {0,1} to N
Please prove a) and b), thank you. + B is a bijection, then (a) (Theorem 8.32)...
Please prove a) and b), thank you.
+ B is a bijection, then (a) (Theorem 8.32) Let A and B be sets such that A is countable. If f: A B is countable. (b) (Theorem 8.33) Every subset of a countable set is countable.1
1. Show that if A and B are countable sets, then AUB is countable. 2. Show...
1. Show that if A and B are countable sets, then AUB is countable. 2. Show that if An are finite sets indexed by positive integers, then Un An is countable. 3. Show that if A and B are countable sets, then A x B is countable. 4. Show that any open set in R is a countable union of open intervals. 5. Show that any function on R can have at most countable many local maximals. Us
all parts A-E please. Problem 8.43. For sake of a contradiction, assume the interval (0,1) is countable. Then there exists a bijection f : N-> (0,1). For each n є N, its image under f is some numb...
all parts A-E please.
Problem 8.43. For sake of a contradiction, assume the interval (0,1) is countable. Then there exists a bijection f : N-> (0,1). For each n є N, its image under f is some number in (0, 1). Let f(n) :-0.aina2na3n , where ain 1s the first digit in the decimal form for the image of n, a2 is the second digit, and so on. If f (n) terminates after k digits, then our convention will be...
4. Let n be a positive integer. Z" is the set of all lists of length n whose entries are in Z. Prove that Z" is...
4. Let n be a positive integer. Z" is the set of all lists of length n whose entries are in Z. Prove that Z" is countable. (Hint: Find a bijection between Z"-1x Z and Z" and then use induction.)
4. Let n be a positive integer. Z" is the set of all lists of length n whose entries are in Z. Prove that Z" is countable. (Hint: Find a bijection between Z"-1x Z and Z" and then use induction.)
(5) Let A, B and C be sets. Show that there is a bijection between the sets F(A, B x C) and F(A, B) x F(A, C) (5) Let A, B and C be sets. Show that there is a bijection between the sets F(A, B x...
(5) Let A, B and C be sets. Show that there is a bijection between the sets F(A, B x C) and F(A, B) x F(A, C)
(5) Let A, B and C be sets. Show that there is a bijection between the sets F(A, B x C) and F(A, B) x F(A, C)
E finite dimensional vector Spaces, dim Show that there is a bijection between the set Bil(V,W;R)...
e finite dimensional vector Spaces, dim Show that there is a bijection between the set Bil(V,W;R) of all bilinear functions on V and W and the set ĢLn(m)R of all matrices of order n x m.
e finite dimensional vector Spaces, dim Show that there is a bijection between the set Bil(V,W;R) of all bilinear functions on V and W and the set ĢLn(m)R of all matrices of order n x m.
12) Let S1 be a countable set, S2 a set that is not countable, and S1...
12) Let S1 be a countable set, S2 a set that is not countable, and S1 ⊂ S2. a) Show that S2 must then contain an infinite number of elements that are not in S1. b) Show that in fact S2 − S1cannot be countable.
Let S be a finite set with cardinality n>0. a. Prove, by constructing a bijection, that the number of subsets of S of size k is equal to the number of subsets of size n- k. Be sure to prove that vour mapping is both injective and surjective. b. Prove, by constructing a bijection, that the number of odd-cardinality subsets of S is equal to the number of even-cardinality subsets of S. Be sure to prove that your mapping is...
2. Determine whether the given sets are countable or uncountable. Justify each answer with a bijection (or table like we did with Q+) or using results from class/textbook. (a) {0, 1, 2} * N (b) A = {(x, y) : x2 + y2 = 1} (c) {0, 1} R Che set of all 2-element subsets of N (e) Real numbers with decimal representations consists of all 1s. (f) The set of all functions from {0,1} to N
Please prove a) and b), thank you.
+ B is a bijection, then (a) (Theorem 8.32) Let A and B be sets such that A is countable. If f: A B is countable. (b) (Theorem 8.33) Every subset of a countable set is countable.1
1. Show that if A and B are countable sets, then AUB is countable. 2. Show that if An are finite sets indexed by positive integers, then Un An is countable. 3. Show that if A and B are countable sets, then A x B is countable. 4. Show that any open set in R is a countable union of open intervals. 5. Show that any function on R can have at most countable many local maximals. Us
all parts A-E please.
Problem 8.43. For sake of a contradiction, assume the interval (0,1) is countable. Then there exists a bijection f : N-> (0,1). For each n є N, its image under f is some number in (0, 1). Let f(n) :-0.aina2na3n , where ain 1s the first digit in the decimal form for the image of n, a2 is the second digit, and so on. If f (n) terminates after k digits, then our convention will be...
4. Let n be a positive integer. Z" is the set of all lists of length n whose entries are in Z. Prove that Z" is countable. (Hint: Find a bijection between Z"-1x Z and Z" and then use induction.)
4. Let n be a positive integer. Z" is the set of all lists of length n whose entries are in Z. Prove that Z" is countable. (Hint: Find a bijection between Z"-1x Z and Z" and then use induction.)
(5) Let A, B and C be sets. Show that there is a bijection between the sets F(A, B x C) and F(A, B) x F(A, C)
(5) Let A, B and C be sets. Show that there is a bijection between the sets F(A, B x C) and F(A, B) x F(A, C)
e finite dimensional vector Spaces, dim Show that there is a bijection between the set Bil(V,W;R) of all bilinear functions on V and W and the set ĢLn(m)R of all matrices of order n x m.
e finite dimensional vector Spaces, dim Show that there is a bijection between the set Bil(V,W;R) of all bilinear functions on V and W and the set ĢLn(m)R of all matrices of order n x m.
Most questions answered within 3 hours.
-
Two capacitors connected in parallel produce an equivalent
capacitance of 45.0 μF but when connected in...
asked 2 minutes ago -
The differences between the two sets of dependent data are -1,
2,-,2, 2. Round to the...
asked 19 minutes ago -
A χ2-curve, looking at the relationship between age and hours
spent working at an office per...
asked 50 minutes ago -
The pH of a sample of water from a river is 5.0. A
sample of effluent from...
asked 1 hour ago -
At the beginning of the period, the Fabricating Department
budgeted direct labor of $136,500 and equipment...
asked 2 hours ago -
Please answer all
____ 28. Rent control is usually
justified on the grounds that it protects...
asked 2 hours ago -
PARTS A-D HAVE BEEN ANSWERED. WAS TOLD TO REPOST. ONLY ANSWER
PARTS E and F.
A...
asked 2 hours ago -
2) You are given the task of finding a representation for a
circle in a drawing...
asked 3 hours ago -
STUDY QUESTION: Does use of diet drug fen-phen
(fenfluramine-phentermine) cause valvular heart disease?
HINT: Valvular heart...
asked 3 hours ago -
1. An object weighing 40 N rests on a surface. The coefficient
of friction is 0.35....
asked 4 hours ago -
Investor company owns 35% of investee company voting stock and
accounts for the investment under the...
asked 5 hours ago -
The number of major faults on a randomly chosen 1 km stretch of
highway has a...
asked 6 hours ago