Question

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 ≤ 1, 0 ≤ y ≤ 1}.

(c) The set [0, 1]Q × [0, 1]R = {(x, y) | x ∈ Q, y ∈ R, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}.

(d) The set N × N × N = {(a, b, c) | a, b, c ∈ N}.

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
Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are...
Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. For those that are finite or uncountable, explain your reasoning. a. integers that are divisible by 7 or divisible by 10
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...
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).
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.)
For each of the following sets X and collections T of open subsets decide whether the...
For each of the following sets X and collections T of open subsets decide whether the pair X, T satisfies the axioms of a topological space. If it does, determine the connected components of X. If it is not a topological space then exhibit one axiom that fails. (a) X = {1, 2, 3, 4} and T = {∅, {1}, {1, 2}, {2, 3}, {1, 2, 3}, {1, 2, 3, 4}}. (b) X = {1, 2, 3, 4} and T...
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)...
For each of the following statements, determine and explain whether it is true or false: a)...
For each of the following statements, determine and explain whether it is true or false: a) If y[n] = x[n] ∗ h[n], then y[n − 1] = x[n − 1] ∗ h[n − 1]. b) If y(t) = x(t) ∗ h(t), then y(−t) = x(−t) ∗ h(−t). c) If x(t) = 0 for t > T1 and h(t) = 0 for t > T2, then x(t) ∗ h(t) = 0 for t > T1 + T2.
(a) Let A and B be countably infinite sets. Decide whether the following are true for...
(a) Let A and B be countably infinite sets. Decide whether the following are true for all, some (but not all), or no such sets, and give reasons for your answers.  A ∪B is countably infinite  A ∩B is countably infinite  A\B is countably infinite, where A ∖ B = { x | x ∈ A ∧ X ∉ B }. (b) Let F be the set of all total unary functions f : N → N...
Decide whether each of the given sets is a group with respect to the indicated operation....
Decide whether each of the given sets is a group with respect to the indicated operation. If it is not a group, state a condition in the definition of group that fails to hold. (a) The set Z+ of all positive integers with operation multiplication. (b) For a fixed integer n, the set of all complex numbers x such that xn = 1 (That is, the set of all nth roots of 1), with operation multiplication. (c) The set Q'...
Find the LUB and GLB of the following sets: (i) {x | x = 2^(−p)+3^(−q )for...
Find the LUB and GLB of the following sets: (i) {x | x = 2^(−p)+3^(−q )for some p,q ∈ N} (ii) {x ∈ R | 3x^(2)−4x < 1} (iii) the set of all real numbers between 0 and 1 whose decimal expression contains no nines
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT