True False or Incoherent 1. A countable union of countable sets is countable. 2. An arbitrary union
Fantastic news! We've Found the answer you've been seeking!
Question:
True False or Incoherent
Transcribed Image Text:
1. A countable union of countable sets is countable. 2. An arbitrary union of countable sets is countable. 3. A finite Cartesian product of countable sets is always countable. 4. A countable Cartesian product of countable sets is countable. 5. All countable infinities are of the same size. 6. All uncountable infinities are of the same size. 7. The cardinality of the power set P(R) is bigger than card (A), where A is the set of all functions f. R → {0, 1}. 8. card(R) < card(C[0, 1]). [This is hard!] 9. Every infinite set has a proper subset of the same cardinality. 10. Cantor's diagonalization argument may be used to show that one uncountably infinite set is bigger than another uncountably infinite set. 11. Every irrational number is a root of some polynomial with integer coefficients. = √d+p. 12. If d and p are metric functions on M, then so is σ = 13. Let A be a subset of the metric space (M, d), then diam(A) = inf {d(x, y); x, y = A}. 14. If R is equipped with the discrete metric, then diam (0, 4) = 4. 15. {1/n) is a Cauchy sequence. 16. Every convergent sequence is a Cauchy sequence. 17. Every Cauchy sequence is convergent. 18. A Cauchy sequence may not be bounded. 19. Every subsequence of a Cauchy sequence is Cauchy. 20. If a Cauchy sequence has a convergent subsequence, then the Cauchy sequence converges. 21. Equivalent metrics preserve Cauchy sequences. That is, if d and p are equivalent on M and {x} is a sequence in M, then {x} is Cauchy under the metric d if and only if {x} is Cauchy under the metric p. 22. An arbitrary union of open sets is open. 23. An infinite intersection of open sets is never open. 24. An arbitrary intersection of closed sets is always closed. 25. A finite union of closed sets is always closed. 26. All sets are either open or closed. 27. If R is equipped with the discrete metric, the set (0, 1) is closed. 1. A countable union of countable sets is countable. 2. An arbitrary union of countable sets is countable. 3. A finite Cartesian product of countable sets is always countable. 4. A countable Cartesian product of countable sets is countable. 5. All countable infinities are of the same size. 6. All uncountable infinities are of the same size. 7. The cardinality of the power set P(R) is bigger than card (A), where A is the set of all functions f. R → {0, 1}. 8. card(R) < card(C[0, 1]). [This is hard!] 9. Every infinite set has a proper subset of the same cardinality. 10. Cantor's diagonalization argument may be used to show that one uncountably infinite set is bigger than another uncountably infinite set. 11. Every irrational number is a root of some polynomial with integer coefficients. = √d+p. 12. If d and p are metric functions on M, then so is σ = 13. Let A be a subset of the metric space (M, d), then diam(A) = inf {d(x, y); x, y = A}. 14. If R is equipped with the discrete metric, then diam (0, 4) = 4. 15. {1/n) is a Cauchy sequence. 16. Every convergent sequence is a Cauchy sequence. 17. Every Cauchy sequence is convergent. 18. A Cauchy sequence may not be bounded. 19. Every subsequence of a Cauchy sequence is Cauchy. 20. If a Cauchy sequence has a convergent subsequence, then the Cauchy sequence converges. 21. Equivalent metrics preserve Cauchy sequences. That is, if d and p are equivalent on M and {x} is a sequence in M, then {x} is Cauchy under the metric d if and only if {x} is Cauchy under the metric p. 22. An arbitrary union of open sets is open. 23. An infinite intersection of open sets is never open. 24. An arbitrary intersection of closed sets is always closed. 25. A finite union of closed sets is always closed. 26. All sets are either open or closed. 27. If R is equipped with the discrete metric, the set (0, 1) is closed.
Expert Answer:
Answer rating: 100% (QA)
The detailed answer for the above question is provided below ANSWER AND STEPBYSTEP EXPLANATION 1 True This is a wellknown result in set theory One can ... View the full answer
Related Book For
Posted Date:
Students also viewed these mathematics questions
-
True / False Questions: If false, give a counterexample or a brief explanation? (a) A solution set of a homogeneous system of linear algebraic equations, given by Is a subspace of R4. True or false?...
-
True or False An annuity is a stream of equal payments that are received or paid at random periods of time?
-
True or False: 1. If the insured makes any false statement in the application process, the insurance policy is voidable. 2. Once an insurance company issues a binder, the policy is irrevocable. 3....
-
Violet Flowers expects to sell 3,000 plants a month. She estimated the following monthly costs: Variable Costs $9,000 Fixed Costs $15,000 During her second month of operation, Violet would like to...
-
Working individually or in groups, research the unemployment rate and laws of your state. Write a summary detailing your states unemployment laws. Assuming Company X has a 30% rate of personnel...
-
Benzene and toluene form nearly ideal solutions. The boiling point of pure benzene is 80.1*C, Calculate the chemical potential of benzene relative to that of pure benzene when xbmzenc = 0.30 at its...
-
Briefly discuss the strengths and limitations associated with this approach and the specific design . Develop a hypothetical research scenario that would necessitate the use of the Action Research...
-
On January 1, Guillen Corporation had 95,000 shares of no-par common stock issued and outstanding. The stock has a stated value of $5 per share. During the year, the following occurred. Apr. 1 Issued...
-
Discuss the effects of all five major accounting assumptions on the accounting process
-
1. What is the cost of debt for Sunrise Bakery? 2. What is their cost of equity? 3. What is the WACC? 4. Which cost of capital should be used to evaluate the feasibility of the oven purchase? 5....
-
Heather and Dave live together. Heather works at a law firm, where she earns $250,000 a year. Dave does most of the work around the house so that Heather can focus all of her time at the law firm....
-
Is tax good or bad? Why do we need to pay tax? Please explain fully and give examples
-
Josh purchased a personal vacation home (Muskoka cottage) through his corporation. Josh uses the cottage to enjoy his vacations with friends and family. The corporation funded the purchase and pays...
-
Describe how social norms within a culture can create stigmas toward sex and gender.
-
Identify and discuss two ways you could seek feedback on the way you handled the intake assessment to allow you to reflect on your own performance. Explain why each option provided is appropriate.
-
What are the multifaceted social variables and determinants that contribute to the emergence and exacerbation of the opioid crisis on a global scale?
-
Suppose that Cana is deciding whether or not to buy a pair of sneakers that she has been researching online, and also the best place to make her purchase. Three different stores in the area sell the...
-
What are the typical record-at-a-time operations for accessing a file? Which of these depend on the current file record?
-
Suppose that is of bounded variation on a closed, bounded interval [a, b]. Prove that is continuous on (a, b) if and only if is uniformly continuous on {a, b).
-
Using Definition 3.35, prove that each of the following functions is uniformly continuous on (0,1). a) f(x) = x2 +x b) f(x) = x3 - x + 2 c) f(*) = x sin 2x
-
Suppose that f: R R is periodic, piecewise continuous, and of bounded variation on R. Prove that if S is a trigonometric series which converges to (f(x+) + f(x-))/2 for all x R, then S is the...
-
An investor is considering adding three new securities to her internationally focused, fixed-income portfolio. She considers the following non-callable securities: 1-year government bond 10-year...
-
Jo Akumbas portfolio is invested in a range of developed markets fixed-income securities. She asks her adviser about the possibility of diversifying her investments to include emerging and frontier...
-
An analyst is reviewing various asset alternatives and is presented with the following information relating to the broad equity market of Switzerland and various industries within the Swiss market...
Study smarter with the SolutionInn App