Question

Match List-I with List-II and choose the correct option:

LIST-I (Set)	LIST-II (Supremum/Infimum)
(A) \( S = \{2, 3, 5, 10\} \)	(III) Sup S = 10, Inf S = 2
(B) \( S = (1, 2] \cup [3, 8) \)	(IV) Sup S = 8, Inf S = 1
(C) \( S = \{2, 2^2, 2^3, \dots, 2^n, \dots\} \)	(II) Sup S = 5, Inf S = -5
(D) \( S = \{x \in \mathbb{Z} : x^2 \le 25\} \)	(I) Inf S = 2

Choose the correct answer from the options given below:

• A. A - I, B - II, C - III, D - IV
• B. A - I, B - III, C - II, D - IV
• C. A - I, B - II, C - IV, D - III
• D. A - III, B - IV, C - I, D - II

Hint:

Remember the key difference: Infimum and Supremum are properties of a set of numbers, while Minimum and Maximum are specific elements. A set can have an infimum/supremum without having a minimum/maximum (e.g., the open interval (0,1) has inf=0 and sup=1, but no min or max).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This question requires finding the infimum (greatest lower bound) and supremum (least upper bound) for four different sets of real numbers.

Step 2: Detailed Explanation:
A. \( S = \{2, 3, 5, 10\} \).
This is a finite set. For a finite set, the infimum is its minimum element and the supremum is its maximum element. \(\inf S = 2\) and \(\sup S = 10\). This matches III. Sup S = 10, Inf S = 2.
B. \( S = (1, 2] \cup [3, 8) \).
The set of lower bounds for S is \((-\infty, 1]\). The greatest lower bound is \(\inf S = 1\). The set of upper bounds for S is \([8, \infty)\). The least upper bound is \(\sup S = 8\). Note that the infimum and supremum do not need to be elements of the set. This matches IV. Sup S = 8, Inf S = 1.
C. \( S = \{2, 2^2, 2^3, ..., 2^n, ...\} = \{2, 4, 8, ...\} \).
This set is not bounded above, so it has no supremum in \(\mathbb{R}\). However, it is bounded below. The smallest element is 2. The set of lower bounds is \((-\infty, 2]\). The greatest lower bound is \(\inf S = 2\). Among the options, I. Inf S = 2 is the only correct statement about this set.
D. \( S = \{x \in \mathbb{Z} : x^2 \le 25\} \).
The inequality \(x^2 \le 25\) is equivalent to \(-5 \le x \le 5\). Since \(x\) must be an integer, the set is \(S = \{-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}\). This is a finite set. The minimum element is -5 and the maximum is 5. \(\inf S = -5\) and \(\sup S = 5\). This matches II. Sup S = 5, Inf S = -5.

Step 3: Final Answer:
The correct pairings are A-III, B-IV, C-I, and D-II. This corresponds to option (D).