Theorems about Countable Sets This handout summarizes some of the most important results about countable sets. (This corollary is just a minor “fussy” step from Theorem 5. Assume A U B is uncountable. 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. (In particular, the union of two countable sets is countable.) In what follows we will often use the correspondence of countable sets and lists (Proposition 14) without further mention. Corollary 6 A union of a finite number of countable sets is countable. A set Ais said to be countably in nite if jAj= jNj, and simply countable if jAj jNj. 11. The Union and Intersection of Two Countable Sets is Countable The Union of Two Countable Sets is Countable This means that there is no function, k: N … 12. THE BASIC TRICHOTOMY: FINITE, COUNTABLE, UNCOUNTABLE PETE L. CLARK 1. Indeed, and a common way to prove it requires you to know that [math]\mathbb{N}\times\mathbb{N}[/math] is countable. Hence T is uncountable. their union Sn k=1 Ak is countable and their Cartesian product A1×A2×... ×An is countable. If T were countable then R would be the union of two countable sets. Otherwise the set A is called infinite. 1) Assume that the real numbers are countable. 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. More About Countable Sets Please read this handout after Section 9.2 in the textbook. their union Sn k=1 Ak is countable and their Cartesian product A1×A2×... ×An is countable. Proposition 16 Let An for n ∈ N be countable sets. The Union and Intersection of Two Countable Sets is Countable The Union of Two Countable Sets is Countable their union Sn k=1 Ak is countable and their Cartesian product A1×A2×... ×An is countable. 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 THE BASIC TRICHOTOMY: FINITE, COUNTABLE, UNCOUNTABLE PETE L. CLARK 1. Proposition 16 Let An for n ∈ N be countable sets. Proposition 16 Let An for n ∈ N be countable sets. Thanks Otherwise the set A is called infinite. Proof by a contradiction. Then S∞ n=1 An is a countable set. Answer to: Prove that the union of countably many countable sets is countable. Then S∞ n=1 An is a countable set. 2) Then every subset of the reals is countable, in particular, the Note that R = A∪ T and A is countable. So we are talking about a countable union of countable sets, which is countable by the previous theorem. Let A denote the set of algebraic numbers and let T denote the set of tran-scendental numbers. A countable union is a union of countably many elements. LetX beanuncountableset. In what follows we will often use the correspondence of countable sets and lists (Proposition 14) without further mention.