- published: 21 May 2010
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In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.
There are many equivalent characterizations of uncountability. A set X is uncountable if and only if any of the following conditions holds:
The first three of these characterizations can be proven equivalent in Zermelo–Fraenkel set theory without the axiom of choice, but the equivalence of the third and fourth cannot be proved without additional choice principles.
Real Analysis, Lecture 7: Countable and Uncountable Sets
Countable and Uncountable Sets (Part 1 of 2)
Countable and Uncountable Sets (Part 2 of 2)
The Cantor Set Is Uncountable
Countable and Uncountable Sets (1/3)
Countable and Uncountable Sets (3/3)
Mod-06 Lec-08 Finite, Infinite, Countable and Uncountable Sets of Real Numbers
Demo Lecture on Countable and Uncountable sets by Shiv(Part 1)
Lecture on countable and uncountable sets
The power set of IN is uncountable
Real Analysis, Spring 2010, Harvey Mudd College, Professor Francis Su. Playlist, FAQ, writing handout, notes available at: http://analysisyawp.blogspot.com/
Part 1 of my video tutorial on countable and uncountable sets. Part 2 is here: http://www.youtube.com/watch?v=fRhdpyaOhEo 00:00 - Introduction 02:43 - Definition of a countable set 03:27 - What does one-to-one correspondence mean? 05:24 - Bijective functions
Part 2 of my video tutorial on countable and uncountable sets. Part 1 is here: http://www.youtube.com/watch?v=sT9hAmaot8U 00:00 - How to show that a set is countable 00:36 - The set of integers is countable 01:19 - The set of rational numbers is countable 02:29 - The set of real numbers is uncountable 05:23 - Summary
Professor Zap Sketches the proof that the Cantor set is uncountable.
Part 1 of 3 From Section 1.3 in Intermediate (Real) Analysis Dr. Scott R. Franklin Go to Part 2 here: http://youtu.be/9XVdTFo2ZCE
A Basic Course in Real Analysis by Prof. P.D. Srivastava, Department of Mathematics, IITKharagpur. For more details on NPTEL visit http://nptel.iitm.ac.in
Demo Lecture on Countable and Uncountable sets by Shiv(Part 1)
This video uses Cantor's diagonal argument to prove that the power set of the natural numbers is uncountable. We first get a feel for why this might be the case, then prove it.
Camera Sponsor: Thomas, Research Scholar (CS), IIT Hyderabad. Recording and editing: Raju M, Research Scholar (A&ME;), IIT Hyderabad.
Cartesian product of uncountable sets is uncountable; Cantor's diagonal process; metric spaces; basic topological notions (limit point, isolated point, closed set, interior point, open set); set is closed iff its complement is open
Recall the complex numbers: the plane with addition and multiplication. Geometric interpretation of operations. Same thing as a+bi. Complex conjugate. Absolute value (modulus) of a complex numbers; properties (esp., triangle inequality). Cauchy-Schwarz inequality. Recall Euclidean space, properties of the norm. Basic topology: one-to-one, onto, finite, countable, uncountable. Every infinite subset of a countable set is countable. A countable union of countable sets is countable.
Real Analysis by Prof. S.H. Kulkarni, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://nptel.iitm.ac.in
Strogatz 11.1 countable vs uncountable
Basic Methods: We continue the study of cardinality with infinite sets. First the class of countably infinite sets is considered, and basic results given. Then we give examples of uncountable sets using Cantor diagonalization arguments.
(Description coming soon!)
"Theory of Computation"; Portland State University: Prof. Harry Porter; www.cs.pdx/~harry
The nineteenth lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers a brief discussion of different sizes of infinity. (The video “Beyond Infinity” may also be helpful here.) Sets with the same cardinality. Countable sets and uncountable sets defined. Connections between functions and sequences. Countability in terms of sequences. The union of two countable sets is countable. (Proved at the start of next class.) These videos are also available for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathematics/id950755120 Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com. Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham.