Question:
Which of the following is a disconnected subset of the real line $\mathbb{R}$?
Which of the following is a disconnected subset of the real line $\mathbb{R}$?
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Key Theorem in Real Analysis: A subset $S \subseteq \mathbb{R}$ is connected if and only if $S$ is an interval. If $S$ is a union of two or more disjoint intervals, it is always disconnected!
Updated On: Jul 29, 2026
- $\{x \mid |x| \leq 2\}$
- $\{x \mid -5 < x \leq 3\}$
- $\{x \mid |x| > 5\}$
- $\{x \mid -5 \leq x < 3\}$
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The Correct Option is C
Solution and Explanation
Step 1: Concept
In standard topology on $\mathbb{R}$, a subset $S \subseteq \mathbb{R}$ is connected if and only if it is an interval (open, closed, half-open, bounded, or unbounded). A subset $S \subseteq \mathbb{R}$ is disconnected if it can be written as $S = A \cup B$ where $A$ and $B$ are non-empty, disjoint sets separated from each other in the subspace topology (i.e., not an interval).
Step 2: Key Formulas and Approach
Translate each set into interval notation and check if it forms a single contiguous interval.
Step 3: Step-by-step Explanation
• Option (A): $\{x \mid |x| \leq 2\ = [-2, 2]$}
This is a single closed bounded interval. Any interval in $\mathbb{R}$ is connected. Thus, Option (A) is connected.
• Option (B): $\{x \mid -5 < x \leq 3\ = (-5, 3]$}
This is a single half-open interval. Hence, Option (B) is connected.
• Option (C): $\{x \mid |x| > 5\ = (-\infty, -5) \cup (5, \infty)$}
This set consists of two separated, disjoint non-empty open sets $A = (-\infty, -5)$ and $B = (5, \infty)$. There is a "gap" between $-5$ and $5$ (e.g., $0 \notin S$). Since it cannot be represented as a single interval, it is disconnected.
• Option (D): $\{x \mid -5 \leq x < 3\ = [-5, 3)$}
This is a single half-open interval, which is connected.
Step 4: Final Answer
The set $\{x \mid |x| > 5\} = (-\infty, -5) \cup (5, \infty)$ is disconnected. Thus, Option (C) is correct.
In standard topology on $\mathbb{R}$, a subset $S \subseteq \mathbb{R}$ is connected if and only if it is an interval (open, closed, half-open, bounded, or unbounded). A subset $S \subseteq \mathbb{R}$ is disconnected if it can be written as $S = A \cup B$ where $A$ and $B$ are non-empty, disjoint sets separated from each other in the subspace topology (i.e., not an interval).
Step 2: Key Formulas and Approach
Translate each set into interval notation and check if it forms a single contiguous interval.
Step 3: Step-by-step Explanation
• Option (A): $\{x \mid |x| \leq 2\ = [-2, 2]$}
This is a single closed bounded interval. Any interval in $\mathbb{R}$ is connected. Thus, Option (A) is connected.
• Option (B): $\{x \mid -5 < x \leq 3\ = (-5, 3]$}
This is a single half-open interval. Hence, Option (B) is connected.
• Option (C): $\{x \mid |x| > 5\ = (-\infty, -5) \cup (5, \infty)$}
This set consists of two separated, disjoint non-empty open sets $A = (-\infty, -5)$ and $B = (5, \infty)$. There is a "gap" between $-5$ and $5$ (e.g., $0 \notin S$). Since it cannot be represented as a single interval, it is disconnected.
• Option (D): $\{x \mid -5 \leq x < 3\ = [-5, 3)$}
This is a single half-open interval, which is connected.
Step 4: Final Answer
The set $\{x \mid |x| > 5\} = (-\infty, -5) \cup (5, \infty)$ is disconnected. Thus, Option (C) is correct.
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