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: Understanding the Concept:
This question tests knowledge of basic topological and set-theoretic properties of common subsets of the real numbers (\(\mathbb{R}\)). We need to determine if each set is open or closed, bounded or unbounded, and countable or uncountable.
Step 2: Detailed Explanation:
Let's analyze each set in List-I and match it with its properties in List-II.
(A) Set of natural numbers, \(\mathbb{N} = \{1, 2, 3, \dots\}\): It is not open because for any natural number n, no open interval around n is contained entirely within \(\mathbb{N}\). It is a closed set in \(\mathbb{R}\). Its complement, \(\mathbb{R} \setminus \mathbb{N}\), is a union of open intervals like \((-\infty, 1) \cup (1, 2) \cup \dots\), which is an open set. This matches with (II) closed.
(B) Open interval (a, b): By its definition in topology, an open interval is an open set. For every point x in (a, b), there exists a smaller open interval around x that is still contained within (a, b). This matches with (I) open.
(C) Set of rational numbers, \(\mathbb{Q}\): It is unbounded both above and below. It is countable, meaning its elements can be put into a one-to-one correspondence with the natural numbers. It is neither open nor closed. The property that fits best is (IV) unbounded below and countable.
(D) Set of irrational numbers, \(\mathbb{Q}^c\): It is unbounded both above and below. It is uncountable. It is neither open nor closed. The property that fits best is (III) unbounded and uncountable.
Step 3: Final Answer:
The correct matching is:
(A) \(\rightarrow\) (II)
(B) \(\rightarrow\) (I)
(C) \(\rightarrow\) (IV)
(D) \(\rightarrow\) (III)
This corresponds to option (3).