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Consider the set A = {x ∈ ℚ : 0 < (√2 - 1)x < √2 + 1} as a subset of ℝ. Which of the following statements is true?

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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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Detailed Solution

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Concept:

The supremum  of the set is the least upper bound for  . The infimum  is the greatest lower bound of 

Explanation:

 

We are given the set } as a subset of , and we need to determine

the correct supremum (sup) and infimum (inf) of this set.

We are asked to solve for    in the inequality .

⇒  2.

Multiply the numerator and denominator by  

So, the inequality becomes 

The set  consists of rational numbers  that satisfy this inequality. Therefore,

The supremum  of this set is  , which is the least upper bound for  .

The infimum  is 0 , since  approaches but never reaches 0 from the positive side.

The correct statement is 

Hence option 2) is correct.

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