Question

(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 and FC the set of all total unary computable functions f : N → N.  Show that the set FC is countably infinite.  Is the set F\FC also countably infinite? Give reasons for your answers. 1 Mark per bullet point.

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

Homework Answers

Answer #1

1) is always countably infinite

can be defined as and where are the bijections from N to A, B respectively

2) is possible and so is so this statement can be true or false

3) can be empty set or A itself so this statement can also be true or false

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
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).
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
1. Let A and B be sets. The set B is of at least the same...
1. Let A and B be sets. The set B is of at least the same size as the set A if and only if (mark all correct answers) there is a bijection from A to B there is a one-to-one function from A to B there is a one-to-one function from B to A there is an onto function from B to A A is a proper subset of B 2. Which of these sets are countable? (mark all...
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'...
Exercise 4.11. For each of the following, state whether it is true or false. If true,...
Exercise 4.11. For each of the following, state whether it is true or false. If true, prove. If false, provide a counterexample. (i) Let X and Y be sets from Rn. If X ⊂ Y then X is closed if and only if Y is closed. (ii) Let X and Y be sets from Rn. If X ∩Y is closed and convex then either X or Y is closed and convex (one or the other). (iii) Let X be an...
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...
Let A, B be sets and f: A -> B. For any subsets X,Y subset of...
Let A, B be sets and f: A -> B. For any subsets X,Y subset of A, X is a subset of Y iff f(x) is a subset of f(Y). Prove your answer. If the statement is false indicate an additional hypothesis the would make the statement true.
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 ≤...
Exercise 4.11. For each of the following, state whether it is true or false. If true,...
Exercise 4.11. For each of the following, state whether it is true or false. If true, prove. If false, provide a counterexample. (i) LetX andY besetsfromRn. IfX⊂Y thenX is closed if and only if Y is closed. (ii) Let X and Y be sets from Rn. If X ∩Y is closed and convex then eitherX or Y is closed and convex (one or the other). (iii) LetX beanopensetandY ⊆X. IfY ≠∅,thenY isaconvexset. (iv) SupposeX isanopensetandY isaconvexset. IfX∩Y ⊂X then X∪Y...
Fill in the blank with “all,” “no,” or “some” to make the following statements true. •...
Fill in the blank with “all,” “no,” or “some” to make the following statements true. • If your answer is “all,” explain why. • If your answer is “no,” give an example and explain. • If your answer is “some,” give two examples, one for which the statement is true and the other for which the statement is false. Explain your examples. 1. For functions g, if lim x→a+ g(x) = 2 and lim x→a− g(x) = −2, then limx→a...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT