countable set example

\end{array}\], \[\left| \mathbb{Q} \right| = {\aleph_0}.\], \[S = \left\{ {{s_1},{s_2},{s_3}, \ldots } \right\}.\], \[{s_1} = {s_{11}}{s_{12}}{s_{13}}{s_{14}}{s_{15}} \cdots\cdot \], \[{s_2} = {s_{21}}{s_{22}}{s_{23}}{s_{24}}{s_{25}} \cdots\cdot \], \[{s_3} = {s_{31}}{s_{32}}{s_{33}}{s_{34}}{s_{35}} \cdots \cdot\], \[\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdot\cdot\], \[{s_n} = {s_{n1}}{s_{n2}}{s_{n3}}{s_{n4}}{s_{n5}} \cdots \], \[{t_i} = \left\{ {\begin{array}{*{20}{l}} To prove part (1), we use a proof by contradiction and assume that A is an infinite set, \(A \thickapprox B\), and \(B\) is not infinite. What Is Meant By Uncountable Set. For example, the even numbers are a countable infinity because you can link the number 2 to the number 1, the number 4 to 2, the number 6 to 3 and so on. For each finite subset C of B, the set of all functions from C to A is just AC, which has cardinality |A||C|, and thus is countable since A is. In set theory, counting is the act of placing things in a one-to-one correspondence with a subset of the natural numbers (not necessarily a proper subset) in such a way that the numbers are used in order with no gaps (each subsequent number is exactly 1 greater than the previous). It's a site that collects all the most frequently asked questions and answers, so you don't have to spend hours on searching anywhere else. Countable sets Definition: A rational number can be expressed as the ratio of two integers p and q such that q 0. Describes a set which contains more elements than the set of integers. Rational numbers (the ratio of two integers such as That is, if we start with a countable set and add nitely many elements, the result is countable. For example, a bag with infinitely many apples would be a countable infinity because (given an infinite amount of time) you can label the apples 1, 2, 3, etc. From our work in Preview Activity \(\PageIndex{2}\), it appears that if n is an even natural number, then \(f(n) > 0\), and if \(n\) is an odd natural number, then \(f(n) \le 0\). Examples of countable set include: Natural Numbers; Even Numbers; Odd Numbers; Whole Numbers; Integers; Positive Integers; Negative Integers, etc. Respectively, the set is called uncountable, if is infinite but | A | | N | , that is, there exists no bijection between the set of natural numbers and the infinite set. We next use those fractions in which the sum of the numerator and denominator is 3. Note that T may or may not be finite. {\left| p \right| + q = 1:} &{\frac{0}{1}}\\[1em] The idea is to use results from Section 9.1 about finite sets to help obtain a contra- diction. Do not delete this text first. Prove that each of the following sets is countably infinite. Use Exercise (9) to prove that if \(A\) and \(B\) are countably infinite sets, then \(A \times B\) is a countably infinite set. If \(A\) is a countably infinite set and \(B\) is a finite set, then \(A \cup B\) is a countably infinite set. A;B countable )A[B, A B countable. The symbol used to represent an empty set is {} or . For example, the set of real numbers is uncountably infinite. So if \(f(m) = f(n)\), then both \(m\) and \(n\) must be even or both \(m\) and \(n\) must be odd. If \(A\) is infinite, let \(f: S \to \mathbb{N}\) be a bijection and define \(g: A \to f(A)\) by \(g(x) = f(x)\), for each \(x \in A\). (Formal proofs are not required.). There are many sets that are countably infinite, , , 2, 3, n, and . One of the main differences between the set of rational numbers and the integers is that given any integer m, there is a next integer, namely \(m + 1\). A set with all the natural numbers (counting numbers) in it is countable too. The cardinality of the set of natural numbers is denoted \(\aleph_0\) (pronounced aleph null): Hence, any countably infinite set has cardinality \(\aleph_0.\). However, we can distinguish infinite from finite sets by using ellipses () Thus the set of rational numbers \(\mathbb{Q}\) is countable, that is. The set of points that remain after all of these intervals are removed is not an interval, however, it is uncountably infinite. Theorem 9.16 says that if we add a finite number of elements to a countably infinite set, the resulting set is still countably infinite. In Part (2) of Preview Activity \(\PageIndex{1}\), we proved that \(D^{+} \thickapprox \mathbb{N}\). This is a contradiction to the assumption that \(A\) is infinite. any subset of a countable set (proof (http://planetmath.org/SubsetsOfCountableSets)). Example 4.3. the set of all cofinite subsets of a countable set. We can write this as a conditional statement as follows: If \(A\) is a finite set, then \(A\) is not equivalent to any of its proper subsets. Definition and Properties of Countable Sets. The Set of Positive Rational Numbers For each i I, there exists a surjection fi: N Ai. For an infinite set to be a . To prove that \(f\) is an injection, we let \(m, n \in \mathbb{N}\) and assume that \(f(m) = f(n)\). Thus the sets \(\mathbb{Z},\) \(\mathbb{O},\) \(\left\{ {a,b,c,d} \right\}\) are countable, but the sets \(\mathbb{R},\) \(\left( {0,1} \right),\) \(\left( {1,\infty } \right)\) are uncountable. Cartesian Product of Countable Sets. A set that is countably infinite is sometimes called a denumerable set. This mapping is bijective. f(0) = n 0:a 00a 01a 02a 03::: 2 Examples of Countable Sets Finite sets are countable sets. Proof. But since X is countable, so is Y. Hence, \(\mathbb{Z}\) is countably infinite and \(\text{card}(\mathbb{Z}) = \aleph_0\). So if we suppose that \(\mathbb{I}\) is countable, then the union of two countable sets \(\mathbb{Q} \cup \mathbb{I} = \mathbb{R}\) would also be countable, which contradicts the above statement. So it is impossible to have a one-to-one relationship between the counting numbers and the real numbers. Following is a summary of some of the main examples dealing with the cardinality of sets that we have explored. For each polynomial p (in one variable X) over , let Rp be the set of roots of p over . Since Q is manifestly infinite, it is countably infinite. For example, we can now conclude that there are infinitely many rational numbers between 0 and \(\dfrac{1}{10000}\) This might suggest that the set \(\mathbb{Q}\) of rational numbers is uncountable. So we will now let \(a\) and \(b\) be any two rational numbers with \(a < b\) and let \(c_1 = \dfrac{a + b}{2}\). It is left as Exercise (6) on page 474 to prove that the function \(h\) is a bijection. Any subset of a countable set is countable. So \(f(1) = \dfrac{1}{1}\). If \(A\) is infinite and \(A \thickapprox B\), then \(B\) is infinite. What appears to be a formula for \(f(n)\) when \(n\) is even? Surprisingly, this is not the case. \[\left| \mathbb{N} \right| = {\aleph_0}.\], \[A = \left\{ {{a_1},{a_2},{a_3}, \ldots ,{a_i}, \ldots } \right\},\;\;B = \left\{ {{b_1},{b_2},{b_3}, \ldots ,{b_j}, \ldots } \right\}.\], \[\left| {\mathbb{N} \times \mathbb{N}} \right| = {\aleph_0}.\], \[\begin{array}{*{20}{l}} The set X of all functions from finite subsets of B into A is countable. . Note that a countable intersection of open sets is not necessarily open. For each \(n \in \mathbb{N}\), the set \(B - \{g(1), g(2), , g(n)\}\) is not empty since \(B\) is infinte. As a result, we get a list of rational numbers that maps to natural numbers. If \(B\) is finte, then \(B\) is countable. Let \(A\) and \(B\) be countably infinite sets and let \(f: \mathbb{N} \to A\) and \(g: \mathbb{N} \to B\) be bijections. In Preview Activity \(\PageIndex{1}\) from Section 9.1, we proved that \(\mathbb{N} \thickapprox D^{+}\), where \(D^{+}\) is the set of all odd natural numbers. (a) Calculate \(f(1)\) through \(f(10)\). A set is uncountable if it contains so many elements that they cannot be put in one-to-one correspondence with the set of natural numbers. Sets such as \(\mathbb{N}\) or \(\mathbb{Z}\) are called countable because we can list their . In the finite set, the process of counting elements comes to an end. So, f maps 0,1,2,3. to The fact that the set of integers is a countably infinite set is important enough to be called a theorem. We first prove that every subset of \(\mathbb{N}\) is countable. Power set of countably finite set is finite and hence countable. fix countable sets A,B. Since Q is manifestly infinite, it is countably infinite. What is \(f(n)\) for \(n\) from 13 to 16? Wow. (A,B,C,D,E denote the vertices of the pentagon.) but in Physics c is the speed of light in a vacuum. Sets are equivalent iff "they are of the same size" (more precisely: iff there is a bijective function between them). A set is called countable, if it is finite or countably infinite. The power set P(A) is defined as a set of all possible subsets of A, including the empty set and the whole set. If \(A\) is a countably infinite set, then \(A \cup \{x\}\) is a countably infinite set. On the right are the integers starting at 0, then 1, then over to -1, on to 2, then -2, etc: The list goes forever and has all the counting numbers and all the integers. Finite sets are sets having a finite/countable number of members. A set X is uncountable if and only if any of the following conditions hold: A set is countably infinite if its elements can be put in one-to-one correspondence with the set of natural numbers. If \(y \le 0\), then \(-2y \ge 0\) and \(1 - 2y\) is an odd natural number. We then see that. Since X is just the union of all AC, where C ranges over the finite subsets of B, and there are countably many of them (as B is countable), X is also countable. Examples of countable sets include the integers, algebraic numbers, and rational numbers. This is the smallest ordinal number after omega. Since an uncountable set is strictly larger than a countable, intuitively this means that an . Corollary 3.9. the set of all finite subsets of a countable set. Are these results consistent with the pattern exhibited at the beginning of this preview activity? In other words, the cardinality of the new set is the same as the cardinality of the original set. We can write all the positive rational numbers in a two-dimensional array as shown in Figure 9.2. We start with all fractions in which the sum of the numerator and denominator is 2 (only \(\dfrac{1}{1}\)). So a countable set can be either finite or countably infinite. Theorem 9.15 is the basis step. Write the contrapositive of the preceding conditional statement. Then their union \(A \cup B\) is also countable. We can then repeat this process to find a rational number between \(\dfrac{5}{12}\) and \(\dfrac{1}{2}\). {0} &{\text{if}\;\;{s_{nn}} = 1}\\ The counting numbers {1, 2, 3, 4, 5, } are countable. We follow the arrows in Figure 9.2 to define \(f: \mathbb{N} \to \mathbb{Q}^{+}\). All examples are both equally interesting and non-interesting for this reason. {1,2,3,4},N,Z,Q are all countable. 12=0.5, 21=2, 9910=9.9, etc) are also countable. Similarly we can show all finite sets are countable. . Since |A| = || and || = ||, then |A| = || = o.. There are two cases: \(A\) is finite or \(A\) is infinite. Informally we can think of this as infinity plus one. By Countable Union of Countable Sets is Countable, it follows that Q is countable. . The answer to this question is yes, but we will wait until the next section to prove that certain sets are uncountable. The basic idea will be to go half way between two rational numbers. We can then conclude that \(a < c_1 < b\). The set of prime numbers less than 10: {2,3,5,7}. Countable set. This page titled 9.2: Countable Sets is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Ted Sundstrom (ScholarWorks @Grand Valley State University) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. The natural numbers is the canonical example of a countably infinite set. By part (c) of Proposition 3.6, the set AB AB is countable. The set of rational numbers \(\mathbb{Q}\) is countable. Just start with $0$, $1$, and an irrational in-between.. The sets N, Z, the set of all odd natural numbers, and the set of all even natural numbers are examples of sets that are countable and countably infinite. The set of positive rational numbers is countably infinite. Georg Cantor showed that the number of real numbers is rigorously larger than a countably infinite set, and the postulate that this number, the so-called " continuum ," is equal to aleph-1 is called the continuum hypothesis. To prove that \(g\) is a surjection, let \(b \in B\) and notice that for some \(k \in \mathbb{N}\), there will be \(k\) natural numbers in \(B\) that are less than \(b\). State whether each of the following is true or false. Note: In Mathematics c is the continuum, . Consider the following set of integers Z: Z = {, -2, -1, 0, 1, 2,} Notation of an Infinite Set: The notation of an infinite set is like any other set with numbers and items enclosed within curly brackets { }. The set of natural numbers, \(\mathbb{N}\), is an infinite set. Although Corollary 9.8 provides one way to prove that a set is infinite, it is sometimes more convenient to use a proof by contradiction to prove that a set is infinite. For example, set S2 representing set of natural numbers is countably infinite. A countable set is either finite or countably infinite.A set that is not countable is called uncountable.. Terminology is not uniform, however: Some authors use "countable" in the . Since \(A \thickapprox B\) and \(B\) is finite, Theorem 9.3 on page 455 implies that \(A\) is a finite set. Thus the set of all rational numbers in is countably infinite and thus countable. By definition, is the union of all Rp, where p ranges over the set P of all polynomials over . Finite vs. Innite Countability Examples Introduction 1. For example, the set of real numbers between 0 and 1 is an uncountable set because no matter what, youll always have at least one number that is not included in the set. We start with a proof that the set of positive rational numbers is countable. In other words, it's called countable if you can put its members into one-to-one correspondence with the natural numbers 1, 2, 3, . We do this by defining the function \(g\) recursively as follows: Every subset of the natural numbers is countable. The symbol \(\aleph\) is the first letter of the Hebrew alphabet, aleph. If the pattern suggested by the function values we have defined continues, what are \(f(11)\) and \(f(12)\)? Modified 3 years, 11 months ago. Derived Examples 1. any finite set, including the empty set (proof ( http:// planetmath .org/AlternativeDefinitionsOfCountable )). Any set that can be arranged in a one-to-one relationship with the counting numbers is also countable. So it seems reasonable to use cases to prove that \(f\) is a surjection and that \(f\) is an injection. . What is countable and uncountable infinite sets? If \(A\) and \(B\) are countable sets, then the Cartesian product \(A \times B\) is also countable. Let's say you list real numbers like this (in some interesting order you chose): But I invent a real number by taking one digit from each number on your list and altering it. You may not say for sure which one (finite or countable) For example, you know that a bijection exists between a set S and a set T, where the set T is a subset of the set of natural numbers. {\left| p \right| + q = 5:} &{\frac{-4}{1}, \frac{-3}{2}, \frac{-2}{3}, \frac{-1}{4}, \frac{1}{4}, \frac{2}{3}, \frac{3}{2}, \frac{4}{1},} as a rational number between \(a\) and \(b\). A countable set is a set of objects that can be counted. On the left are the counting numbers. We continue in this fashion. For example, set S2 representing set of natural numbers is countably infinite. The basic examples of (finite) countable sets are sets given by a list of their elements: The set of even prime numbers that contains only one element: {2}. There is no injective function (hence no bijection) from X to the set of natural numbers. In fact, if \(A = \{a_1, a_2, a_3, \}\) and \(B = \{b_1, b_2, b_3, \}\), then we can use the following diagram to help define a bijection from \(\mathbb{N}\) to \(A \cup B\). Any infinite subset of a countably infinite set is countably infinite. For example, if we use \(a = \dfrac{1}{3}\) and \(b = \dfrac{1}{2}\), we can use, \(\dfrac{a + b}{2} = \dfrac{1}{2} (\dfrac{1}{3} + \dfrac{1}{2}) = \dfrac{5}{12}\). Since \(\mathbb{Q}^{+}\) is countable, it seems reasonable to expect that \(Q\) is countable. Thus, we need to distinguish between two types of infinite sets. For example, the set of positive even numbers is a countable set because each and every term in that set can be matched to a term in the natural numbers set. 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