Problem 1. Let A C R be a countable set. Prove that R\ A is uncountable.
Explain or prove your answer. Is the following set finite, countable or uncountable? {(x, y) E NXR : xy = 1}
Let X be a set and let T be the family of subsets U of X such that X\U (the complement of U) is at most countable, together with the empty set. a) Prove that T is a topology for X. b) Describe the convergent sequences in X with respect to this topology. Prove that if X is uncountable, then there is a subset S of X whose closure contains points that are not limits of the sequences in S....
From the class Introduction to Abstract Algebra on the section of countable and uncountable sets 3. Let X and Y be two nonempty finite sets. Let F(X, Y) denote the set of all function from X to Y. Is this set finite, countably infinite, or uncountable? Prove your answer
please explain it step by step( not use the example with number) thanks 1. Determine whether each of these sets is countable or uncountable. For those that are countably infinite, prove that the set is countably infinite. (a) integers not divisible by 3. (b) integers divisible by 5 but not 7 c: i.he mal ilullilbers with1 € lex"Juual reprtrainiatious" Du:"INǐ lli!", of all is. d) the real numbers with decimal representations of all 1s or 9s. 1. Determine whether each...
Problem 23.7. Prove that a set A is uncountable if there is an injective function
2. (Countable and uncountable sets and diagonalization) (a) A polynomial in variable x is an expression of the form c0 + c1x + c2x2 +c3x3 +· · ·+cdxd , where d is a non-negative integer and c0, · · · , cd are constants, called coefficients. Let P be the set of polynomials with integer coefficients. Show that P is countable.
Show that A is an uncountable set of reals. Let B be the set of reals r that divide A into two uncountable sets; that is, the numbers in A less than r are uncountable, as are those greater than r. Show that B is non-empty. *Please go step by step. Don't skip anything!*
Exercise 3.2.12. Let A be an uncountable set and let B be the set of real numbers that divides A into two uncountable sets; that is, s E B if both {{ : 2 € A and r < s} and {x : x € A and x > s} are uncountable. Show B is nonempty and open. T
3. Let E E Lm* (Lebesgue measurable set). Prove that there exist a set G (a countable intersection of open sets), and a set F (a countable union of closed sets) such that F CE C G and m* (F) the Lebesgue measure of a set Hint: The Lebesgue measure can be calculated in terms of open and closed sets m* (E) m* (G), where m* denotes 3. Let E E Lm* (Lebesgue measurable set). Prove that there exist a...
Problem 6 Suppose A and B are countable sets. Prove A × B is a countable set.