Question

Match List-I with List-II 

\[\begin{array}{|c|c|}\hline \textbf{List-I} & \textbf{List-II} \\ \hline \text{(A) Closed Interval} & (I)\ [a, b] = \{\,x \in \mathbb{R} : a \leq x \leq b\,\} \\ \hline \text{(B) Open Interval} & (II)\ (a, b) = \{\,x \in \mathbb{R} : a < x < b\,\} \\ \hline \text{(C) Unbounded Interval} & (III)\ [a, b) = \{\,x \in \mathbb{R} : a \leq x < b\,\} \\ \hline \text{(D) Half Open Interval} & (IV)\ (a, \infty) = \{\,x \in \mathbb{R} : a < x\,\} \\ \hline \end{array}\]


 

Choose the correct answer from the options given below:
 

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

Hint:

In interval notation, square brackets \([]\) include the endpoints, while parentheses \(()\) exclude them.

Answer 1

The Correct Option is B

Step 1: Understanding the intervals.
- A **Closed Interval** includes both endpoints, so its representation is \([a, b] = \{x \in \mathbb{R}: a \leq x \leq b\}\). This corresponds to option (I).
- An **Open Interval** excludes both endpoints, so its representation is \((a, b) = \{x \in \mathbb{R}: a < x < b\}\). This corresponds to option (II).
- A **Unbounded Interval** is one where one endpoint is infinite, so its representation is \([a, b) = \{x \in \mathbb{R}: a \leq x < b\}\). This corresponds to option (III).
- A **Half Open Interval** includes one endpoint and excludes the other, so its representation is \((a, \infty) = \{x \in \mathbb{R}: a < x\}\). This corresponds to option (IV).

Step 2: Conclusion.
The correct match is (A) - (I), (B) - (II), (C) - (III), (D) - (IV).