Question

Match List-I with List-II 

Choose the correct answer from the options given below: 
 

• (A) - (I), (B) - (II), (C) - (III), (D) - (IV)
• (A) - (I), (B) - (III), (C) - (II), (D) - (IV)
• (A) - (II), (B) - (I), (C) - (IV), (D) - (III)
• (A) - (III), (B) - (IV), (C) - (I), (D) - (II)

Hint:

Key properties to memorize: \(\mathbb{N}\) and \(\mathbb{Z}\) are closed and countable. \(\mathbb{Q}\) is countable, but neither open nor closed. \(\mathbb{Q}^c\) and \(\mathbb{R}\) are uncountable. An open interval (a,b) is open. A closed interval [a,b] is closed.

Answer 1

The Correct Option is C

Step 1: Overview: This question assesses understanding of fundamental topological and set-theoretic characteristics of real number subsets (\(\mathbb{R}\)). The task is to classify each set as open or closed, bounded or unbounded, and countable or uncountable. Step 2: Set Analysis: Each set in List-I is now matched with its corresponding properties from List-II. (A) Natural Numbers, \(\mathbb{N} = \{1, 2, 3, \dots\}\): Not open; no open interval around a natural number is entirely within \(\mathbb{N}\). Closed in \(\mathbb{R}\). The complement, \(\mathbb{R} \setminus \mathbb{N}\), is a union of open intervals, hence open. Matches (II) closed. (B) Open interval (a, b): By definition, an open interval is an open set. For every x in (a, b), a smaller open interval around x exists within (a, b). Matches (I) open. (C) Rational Numbers, \(\mathbb{Q}\): Unbounded both above and below. Countable; elements can be put in one-to-one correspondence with natural numbers. Neither open nor closed. Best match: (IV) unbounded below and countable. (D) Irrational Numbers, \(\mathbb{Q}^c\): Unbounded both above and below. Uncountable. Neither open nor closed. Best match: (III) unbounded and uncountable. Step 3: Solution: The matching is: (A) \(\rightarrow\) (II) (B) \(\rightarrow\) (I) (C) \(\rightarrow\) (IV) (D) \(\rightarrow\) (III) This corresponds to option (3).