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मान लीजिए कि समुच्चय A = {x ∈ ℚ : 0 < (√2 - 1)x < √2 + 1}, ℝ का एक उपसमुच्चय है। निम्नलिखित में से कौन सा कथन सत्य है?

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  1. sup A = 2 + 2√3
  2. sup A = 3 + 2√2
  3. inf A = 2 + 2√3
  4. inf A = 3 + 2√2

Answer (Detailed Solution Below)

Option 2 : sup A = 3 + 2√2
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अवधारणा:

समुच्चय का उच्चक ,  का निम्नतम उपरि परिबंध है। निम्नक ,  का महत्तम निम्नतम परिबंध है।

 

स्पष्टीकरण:


हमें के उपसमुच्चय के रूप में } समुच्चय दिया गया है, और हमें निर्धारित करने की आवश्यकता है

इस समुच्चय का सही उच्चक (sup) और निम्नक (inf)।

हमें असमिका  में का हल निकालने के लिए कहा गया है।

⇒ 2.

अंश और हर को से गुणा करें

इसलिए, असमिका  हो जाती है

समुच्चय  में परिमेय संख्याएँ शामिल हैं जो इस असमिका को संतुष्ट करती हैं। इसलिए,

इस समुच्चय का उच्चक  है, जो के लिए निम्नतम उपरि परिबंध है।

निम्नक , 0 है, क्योंकि धनात्मक पक्ष से 0 के निकट पहुँचता है परंतु कभी 0 तक नहीं पहुँचता है।

सही कथन है। 

अतः विकल्प 2) सही है।

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