सिद्ध कीजिए कि वास्तविक संख्याओं का समुच्चय R अगणनीय है
Answers
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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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
Learn More:
वास्तविक संख्याओं का समुच्चय R अगणनीय है
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