Math, asked by ychawda59, 4 hours ago

सिद्ध कीजिए कि वास्तविक संख्याओं का समुच्चय R अगणनीय है​

Answers

Answered by pd143577
1

Answer:

सिद्ध कीजिए कि वास्तविक संख्याओं के समुच्चय R में R = { (a,b) : a lt b^(2)}, द्वारा परिभाषित संबंध R, न तो स्वतुल्य है, न सममित हैं और न ही संक्रामक है ।

Step-by-step explanation:

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Answered by amitnrw
3

Given :   वास्तविक संख्याओं का समुच्चय R

Set of Real numbers

To Find : सिद्ध कीजिए कि वास्तविक संख्याओं का समुच्चय R अगणनीय है​

Prove that  Set of Real numbers is uncountable

Solution:

Assume that  Set of Real numbers   is countable

and arrange then in ascending  order

R = {  a₁  , a₂  ,  a₃ , _____, n₁ , n₂ ____   z₁ , z₂   }

Each number mentioned above representing a real numbers and arranged in ascending order

As we know that there exist at least a real number between 2 real numbers a , b  is  (a  + b)/2  

a < (  a + b) /2  <   b  if  a < b

a  > ( a + b)/2 > b   if  a > b

Hence there will exist a real number (z₁ + z₂ )/2

z₁ <  (z₁ + z₂ )/2 < z₂

but as we have arranged numbers in ascending order

hence there can not exist any real number between z₁ and   z₂

but (z₁ + z₂ )/2  lies between  z₁ and   z₂

Hence our assumption was wrong

Assumption that  Set of Real numbers   is countable  is  INCORRECT

hence  Set of Real numbers   is uncountable

वास्तविक संख्याओं का समुच्चय R अगणनीय है​

QED

Hence proved

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वास्तविक संख्याओं का समुच्चय R  अगणनीय है​

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