Countably Infinite Set, Countable & Uncountable Set, Cantor’s Diagonal Process | Countability : 03

REAL ANALYSIS (COUNTABILITY OF SETS)
In this video we will discuss Cantor's Theorem with proof.
Countability of Sets | Similar Sets, Finite Sets, Infinite Sets | Real Analysis : lec-01
• Countability of Sets |...
Cantor's Theorem with proof | Countability of Sets | Real Analysis : lec-02
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[@shriyasethi9559]

Hello sir.. I have no words to thankyou enough.. I am also a teacher but seeing you working so hard gives me all the more motivation to work hard.. P.s. -These lectures are savior for distance learning students.. More power to you.. Regards

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Thanku so much sir....

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0

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Have not seen anyone teaching so good. Nicely explained.

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[@purv]

Kitne simple way me countable infinite set smjha Diya sir aapne..🤔 pahli baar achhe se smjh aaya,🙏

[@praveenkumarbrathod7299]

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Sir अच्छी तरह से समझ में आ गया thank you sir

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[@undercherries4659]

What if countability is inconsistent? The following is adopted from Mueckenheim: If the fractions m/n are enumerated by the natural numbers k according to Cantor's function k = (m + n - 1)(m + n - 2)/2 + m then all the fractions of the sequence 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ... are enumerated. But if the natural numbers first are in bijection with the integer fractions of the first column of the matrix 1/1, 1/2, 1/3, 1/4, ... 2/1, 2/2, 2/3, 2/4, ... 3/1, 3/2, 3/3, 3/4, ... 4/1, 4/2, 4/3, 4/4, ... 5/1, 5/2, 5/3, 5/4, ... ... then they must be distributed over the matrix such that no fraction remains without index. That means, there is a permutation such that the X of the first column XOOOO... XOOOO... XOOOO... XOOOO... XOOOO... ... by being exchanged with the O's cover all matrix positions. All O's will vanish. This is obviously impossible because exchanging cannot reduce them. The number of not indexed fractions, represented by O's, will remain constant forever, in infinity.

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