(i) The set of infinite sequences in \(\{1,2,\cdots, b-1\}^{\mathbb{N}}\) is uncountable. (ii) The set of finite sequences (but without bound) in \(\{1, 2, \cdots, b-1\}^{\mathbb{N}}\) is countable. Proof

The purpose of this handout is to explain the notions of countable and uncountable sets. map f between sets S1 and S2 is called a bijection if f is one-to-one and onto. In other words. If f(a) = f(b) then a = b. This holds for all a, b ∈ S1. For each b ∈ …

Definition: A set that is either finite or has the same cardinality as the set of positive integers Z+ is called countable. A set that is not countable is called uncountable. Why these are called countable? • The elements of the set can be enumerated and listed. • Assume A = {0, 2, 4, 6, ... } set of even numbers. Is it countable?

MATH1050 Countable sets and uncountable sets 1. Definition. Let A be a set. (1) A is countable if A.N. (2) A is said to be countably infiniteif A∼N. (3) A is said to be uncountable if A is not countable. Basic examples of countably infinite sets. (a) N,Z,Q; (b) N2, N3, N4, ... . Basic examples of uncountable sets. (a) Map(N,{0,1}), Map(N,J0 ...

14.2-4: Prove: The set RrQ of irrational numbers is uncountable. Let’s try a proof by contradiction: Proof. Suppose RrQ is countable. Then R, as the union R = (RrQ) [Q of the countable sets R r Q and Q, is countable. This contradicts R being uncountable. That worked quite easily, given the theorems we have from the lesson summary.

Two sets A and B are called equinumerous, written A ∼ B, if there is a bijection. : X → Y . A set A is called countably infinite if A ∼ N. We say that A is countable if A ∼ N or A is finite. Example 3.1. The sets (0, ∞) and R are equinumerous. Indeed, the func-tion f : R → (0, ∞) defined by f(x) = ex is a bijection. Example 3.2.

We can count the elements of a countable set one at a time. The objects are “discrete” (in contrast to “continuous”). Discrete mathematics deals with all kinds of countable sets. even = {n | n ∈ N0 and n is even} Obviously: even ⊂ N0 Intuitively, there are twice as many natural numbers as even numbers — no? Is |even| < |N0|?

Countable and Uncountable Definition : If a set is finite, or if it has cardinality א 0 , then we say the set is countable Else, the set is uncountable Georg Cantor in 1891 showed the following amazing result : The set of real numbers is uncountable 32

Subsets of Countable Sets are Countable In general: Theorem (subsets of countable sets are countable) Let A be a countable set. Every set B with B A is countable. Proof. Since A is countable there is an injective function f from A to N 0. The restriction of f to B is an injective function from B to N 0.

We can count the elements of a countable set one at a time. The objects are “discrete” (in contrast to “continuous”). Discrete mathematics deals with all kinds of countable sets. is countably infinite if |A| = |N0|. is countable if |A| ≤ |N0|. set is countable if it is finite or countably infinite.