Proposition 16 Let An for n ∈ N be countable sets. A set Ais said to be countably in nite if jAj= jNj, and simply countable if jAj jNj. In what follows we will often use the correspondence of countable sets and lists (Proposition 14) without further mention. 1) Assume that the real numbers are countable. Say we have a family of sets {eq}\{ A_i ~|~ i \in I \} {/eq} This family is countable if {eq}I {/eq} is countable. 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. Let A denote the set of algebraic numbers and let T denote the set of tran-scendental numbers. Argue that the set of all computer programs is a countable set, but the set of all functions is an uncountable set. Since R is un-countable, R is not the union of two countable sets. their union Sn k=1 Ak is countable and their Cartesian product A1×A2×... ×An is countable. Thanks In words, a set is countable if it has the same cardinality as some subset of the natural numbers. 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. Hence T is uncountable. This means that there is no function, k: N … Indeed, and a common way to prove it requires you to know that [math]\mathbb{N}\times\mathbb{N}[/math] is countable. their union Sn k=1 Ak is countable and their Cartesian product A1×A2×... ×An is countable. A countable union is a union of countably many elements. (In particular, the union of two countable sets is countable.) 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 Union and Intersection of Two Countable Sets is Countable The Union of Two Countable Sets 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 So we are talking about a countable union of countable sets, which is countable by the previous theorem. 8 CS 441 Discrete mathematics for CS M. Hauskrecht Cardinality Theorem: The set of real numbers (R) is an uncountable set. Since R is un-countable, R is not the union of two countable sets. Many of these are proved either in the textbook or in its exercises, but I … 1) Assume that the real numbers are countable. Note that R = A∪ T and A is countable. 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. 3 Countability De nition 3.1. Proof by a contradiction. Otherwise the set A is called infinite. Then S∞ n=1 An is a countable set. In other words, A is a countable set, and every member of A is also a countable set. 13. Corollary 6 A union of a finite number of countable sets is countable. their union Sn k=1 Ak is countable and their Cartesian product A1×A2×... ×An is countable. Let A denote the set of algebraic numbers and let T denote the set of tran-scendental numbers. Proof by a contradiction. Hence T is uncountable. If T were countable then R would be the union of two countable sets. 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. 12. Theorem: The set of all finite-length sequences of natural numbers is countable.