The cardinality of a set is n (A) = x, where x is the number of elements of a set A. \(\aleph_2=|\scr{P}(\mathbb{R})|= |\scr{P}(\scr{P}(\mathbb{N}))|\) cardinality of the next uncountably infinite sets. The set of integers is an infinite set. Example: B = Set of all natural numbers. An infinite set that can be put into a one-to-one correspondence with \(\mathbb{N}\) is countably infinite. If the elements of a set cannot be counted, i.e. Let \(n_i\) be the \(i\)th smallest index such that \(x_{n_i} \in A\). For example, you can write $\mathbb{N}=\{1,2,3,\cdots\}$, $\mathbb{Z}=\{0,1,-1,2,-2,3,-3,\cdots\}$. Since, set A has 5 elements. We can show the set of real numbers in the interval \((0,1)\) are uncountable as follows: is said to be an Infinite Set. Legal. Transfinite numbers are used to describe the cardinalities of "higher & higher" infinities. A set which is not finite is called an infinite set. Write this (infinite) list, and as it's written, we will create a number that is NOT on that list. So, for the second number on the list, we see the second digit is a 5, and we choose a 0 for the second digit of our number being created. \[ \mathbb{N}=\{1,2,3,4,...\}\mbox{ is the set of Natural Numbers, also known as the Counting Numbers}.\], \(\mathbb{N}\) is an infinite set and is the same as \( \mathbb{Z}^+.\). if elements are infinite, it is an infinite set. Other strange math can be done with transfinite numbers such as \(\aleph_1 + \aleph_0 = \aleph_1.\). An infinite set that cannot be put into a one-to-one correspondence with \(\mathbb{N}\) is uncountably infinite. On signing up you are confirming that you have read and agree to Since we cannot count all the natural numbers, Set B is infinite set. 1. \[2 \rightarrow 1 \qquad \qquad 4 \rightarrow 2 \qquad \qquad 6 \rightarrow 3 \qquad \qquad 8 \rightarrow 4 \qquad \mbox{ etc. He has been teaching from the past 9 years. Consider the case that \( A \) is an infinite subset of \( S\). (We choose a 0 unless the digit we are comparing to is a 0 and then we choose a 1.) The continuum hypothesis actually started out as the continuum conjecture, until it was shown to be consistent with the usual axioms of the real number system (by Kurt Gödel in 1940), and independent of those axioms (by Paul Cohen in 1963). \(0.a_{31}a_{32}a_{33}a_{34}a_{35} \ldots \) In this section, we will see how the the Natural Numbers are used as a standard to test if an infinite set is "countably infinite". If set \(A\) and set \(B\) have the same cardinality, then there is a one-to-one correspondence from set \(A\) to set \(B\). by } f(n)=\frac{n-2}{2}.\] \(\mathbb{R}\) is an uncountably infinite set. \(0.a_{21}a_{22}a_{23}a_{24}a_{25} \ldots \) 3. He provides courses for Maths and Science at Teachoo. Instead, we use the more appropriate set-builder notation which describes what elements are contained in the set. It can be shown that this function is well-defined and a bijection.). B = {1, 2, 3, 4, 5, 6, 7, 8, 9, ………. } Examples of Infinite Sets. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. So, even though \(E \subset \mathbb{Z},\) \(|E|=|\mathbb{Z}|.\) Have questions or comments? etc. Singleton set. Consider sets \(P\) and \(Q\). Teachoo is free. Infinite Set. 1) The set of all positive even numbers. \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), [ "article:topic", "authorname:hkwong", "license:ccbyncsa", "showtoc:yes" ], \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\).

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