Question

Prove for each of the following: a. Exercise A union of finitely many or countably many...

Prove for each of the following:

a. Exercise A union of finitely many or countably many countable sets is countable. (Hint: Similar)

b. Theorem: (Cantor 1874, 1891) R is uncountable.

c. Theorem: We write |R| = c the “continuum”. Then c = |P(N)| = 2א0

d. Prove the set I of irrational number is uncountable. (Hint: Contradiction.)

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
a) Prove that the union between two countably infinite sets is a countably infinite set. b)...
a) Prove that the union between two countably infinite sets is a countably infinite set. b) Would the statement above hold if we instead started with an infinite amount of countably infinite sets? _________________________________________________ Thank you in advance!
Problem 3 Countable and Uncountable Sets (a) Show that there are uncountably infinite many real numbers...
Problem 3 Countable and Uncountable Sets (a) Show that there are uncountably infinite many real numbers in the interval (0, 1). (Hint: Prove this by contradiction. Specifically, (i) assume that there are countably infinite real numbers in (0, 1) and denote them as x1, x2, x3, · · · ; (ii) express each real number x1 between 0 and 1 in decimal expansion; (iii) construct a number y whose digits are either 1 or 2. Can you find a way...
Prove that a countable union of countable sets countable; i.e., if {Ai}i∈I is a collection of...
Prove that a countable union of countable sets countable; i.e., if {Ai}i∈I is a collection of sets, indexed by I ⊂ N, with each Ai countable, then union i∈I Ai is countable. Hints: (i) Show that it suffices to prove this for the case in which I = N and, for every i ∈ N, the set Ai is nonempty. (ii) In the case above, a result proven in class shows that for each i ∈ N there is a...
Which of the following sets are finite? countably infinite? uncountable? Give reasons for your answers for...
Which of the following sets are finite? countably infinite? uncountable? Give reasons for your answers for each of the following: (a) {1\n :n ∈ Z\{0}}; (b)R\N; (c){x ∈ N:|x−7|>|x|}; (d)2Z×3Z Please answer questions in clear hand-writing and show me the full process, thank you (Sometimes I get the answer which was difficult to read).
For each of the following sets, determine whether they are countable or uncountable (explain your reasoning)....
For each of the following sets, determine whether they are countable or uncountable (explain your reasoning). For countable sets, provide some explicit counting scheme and list the first 20 elements according to your scheme. (a) The set [0, 1]R × [0, 1]R = {(x, y) | x, y ∈ R, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}. (b) The set [0, 1]Q × [0, 1]Q = {(x, y) | x, y ∈ Q, 0 ≤ x ≤...
Using field axioms and order axioms prove the following theorems (i) The sets R (real numbers),...
Using field axioms and order axioms prove the following theorems (i) The sets R (real numbers), P (positive numbers) and [1, infinity) are all inductive (ii) N (set of natural numbers) is inductive. In particular, 1 is a natural number (iii) If n is a natural number, then n >= 1 (iv) (The induction principle). If M is a subset of N (set of natural numbers) then M = N The following definitions are given: A subset S of R...
1. Write the following sets in list form. (For example, {x | x ∈N,1 ≤ x...
1. Write the following sets in list form. (For example, {x | x ∈N,1 ≤ x < 6} would be {1,2,3,4,5}.) (a) {a | a ∈Z,a2 ≤ 1}. (b) {b2 | b ∈Z,−2 ≤ b ≤ 2} (c) {c | c2 −4c−5 = 0}. (d) {d | d ∈R,d2 < 0}. 2. Let S be the set {1,2,{1,3},{2}}. Answer true or false: (a) 1 ∈ S. (b) {2}⊆ S. (c) 3 ∈ S. (d) {1,3}∈ S. (e) {1,2}∈ S (f)...
Write a program of wordSearch puzzle that use the following text file as an input. The...
Write a program of wordSearch puzzle that use the following text file as an input. The output should be like this: PIXEL found (left) at (0,9). ( Use JAVA Array ) .Please do not use arrylist and the likes! Hints • The puzzle can be represented as a right-sized two-dimensional array of characters (char). • A String can be converted into a right-sized array of characters via the String method toCharArray. . A word can occur in any of 8...
Relations and Functions Usual symbols for the above are; Relations: R1, R2, S, T, etc Functions:...
Relations and Functions Usual symbols for the above are; Relations: R1, R2, S, T, etc Functions: f, g, h, etc. But remember a function is a special kind of relation so it might turn out that a Relation, R, is a function, too. Relations To understand the symbolism better, let’s say the domain of a relation, R, is A = { a, b , c} and the Codomain is B = { 1,2,3,4}. Here is the relation: a R 1,    ...
1. Vim commands: a. How do you auto indent your program? b. Explain what the following...
1. Vim commands: a. How do you auto indent your program? b. Explain what the following commands do: dd, y3, p, :set cindent (1 pt) VIM exercises These exercises on the computer need to be repeated by each student in the pair. This is to ensure that both students understand how to get around in Linux!!! For this part of the lab, you will create a .vimrc file that will help you develop your C++ programs using VIM. First, we...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT