2) Then every subset of the reals is countable, in particular, the But, since others have already shown that method, I’ll show a slightly different proof. In other words, A is a countable set, and every member of A is also a countable set. In what follows we will often use the correspondence of countable sets and lists (Proposition 14) without further mention. (In particular, the union of two countable sets is countable.) Many of these are proved either in the textbook or in its exercises, but I … This means that there is no function, k: N … Proof by a contradiction. Introducing equivalence of sets, countable and uncountable sets We assume known the set Z+ of positive integers, and the set N= Z+ [ f0g of natural numbers. Indeed, and a common way to prove it requires you to know that [math]\mathbb{N}\times\mathbb{N}[/math] is countable. This set is the union of the length-1 sequences, the length-2 sequences, the length-3 sequences, each of which is a countable set (finite Cartesian product). Since R is un-countable, R is not the union of two countable sets. If T were countable then R would be the union of two countable sets. their union Sn k=1 Ak is countable and their Cartesian product A1×A2×... ×An is countable. their union Sn k=1 Ak is countable and their Cartesian product A1×A2×... ×An is countable. 1) Assume that the real numbers are countable. Then S∞ n=1 An is a countable set. their union Sn k=1 Ak is countable and their Cartesian product A1×A2×... ×An is countable. Thanks Otherwise the set A is called infinite. THE BASIC TRICHOTOMY: FINITE, COUNTABLE, UNCOUNTABLE PETE L. CLARK 1. But, since others have already shown that method, I’ll show a slightly different proof. Let A denote the set of algebraic numbers and let T denote the set of tran-scendental numbers. Corollary 6 A union of a finite number of countable sets is countable. Hi, i know that it is consistent with ZF without choice that the reals are the countable union of countable sets. THE BASIC TRICHOTOMY: FINITE, COUNTABLE, UNCOUNTABLE PETE L. CLARK 1. LetX beanuncountableset. So we are talking about a countable union of countable sets, which is countable by the previous theorem. 3 Countable and Uncountable Sets A set A is said to be finite, if A is empty or there is n ∈ N and there is a bijection f : {1,...,n} → A. Let B be the union of all the members of A. Introducing equivalence of sets, countable and uncountable sets We assume known the set Z+ of positive integers, and the set N= Z+ [ f0g of natural numbers. In words, a set is countable if it has the same cardinality as some subset of the natural numbers. Argue that the set of all computer programs is a countable set, but the set of all functions is an uncountable set. Note that R = A∪ T and A is countable. If T were countable then R would be the union of two countable sets. Provethatifthereisa1-1correspondencef:A → X, thenA isuncountable. We must prove that B is a countable set. 3 Countable and Uncountable Sets A set A is said to be finite, if A is empty or there is n ∈ N and there is a bijection f : {1,...,n} → A. Answer to: Prove that the union of countably many countable sets is countable. Hence T is uncountable. Proposition 16 Let An for n ∈ N be countable sets. (Since each program computes a function, this means theremustbethingsitisn’tpossibletowriteaprogramtodo.) Hence T is uncountable. 2) Then every subset of the reals is countable, in particular, the 8 CS 441 Discrete mathematics for CS M. Hauskrecht Cardinality Theorem: The set of real numbers (R) is an uncountable set. (This corollary is just a minor “fussy” step from Theorem 5. Then S∞ n=1 An is a countable set. The Union and Intersection of Two Countable Sets is Countable The Union of Two Countable Sets is Countable Is there any good reference to read a proof? Theorem: The set of all finite-length sequences of natural numbers is countable. Let A denote the set of algebraic numbers and let T denote the set of tran-scendental numbers. 13. Since A and B are countable, there exists f: N -> A and g: N -> B where N is the set of all natural numbers. THE BASIC TRICHOTOMY: FINITE, COUNTABLE, UNCOUNTABLE PETE L. CLARK 1. Note that R = A∪ T and A is countable. 1) Assume that the real numbers are countable. A set Ais said to be countably in nite if jAj= jNj, and simply countable if jAj jNj. Proof by a contradiction. Since R is un-countable, R is not the union of two countable sets. More About Countable Sets Please read this handout after Section 9.2 in the textbook. A countable union is a union of countably many elements. In what follows we will often use the correspondence of countable sets and lists (Proposition 14) without further mention. Otherwise the set A is called infinite. The Union and Intersection of Two Countable Sets is Countable The Union of Two Countable Sets is Countable LetC Proposition 16 Let An for n ∈ N be countable sets. Proposition 16 Let An for n ∈ N be countable sets. Theorems about Countable Sets This handout summarizes some of the most important results about countable sets. 8 CS 441 Discrete mathematics for CS M. Hauskrecht Cardinality Theorem: The set of real numbers (R) is an uncountable set. Indeed, and a common way to prove it requires you to know that [math]\mathbb{N}\times\mathbb{N}[/math] is countable. 11. The way Theorem 5 is stated, it applies to an infinite collection of countable sets If we have only finitely many,E ßÞÞÞßE ßÞÞÞ"8