Question:
Which of the following subsets of \( \mathbb{R} \) is (are) connected?
Which of the following subsets of \( \mathbb{R} \) is (are) connected?
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For connected sets, look for intervals or regions without breaks or gaps. Disjoint sets are not connected.
Updated On: Dec 15, 2025
- \( \{ x \in \mathbb{R} | x^2 + x > 4 \} \)
- \( \{ x \in \mathbb{R} | x^2 + x < 4 \} \)
- \( \{ x \in \mathbb{R} | |x| < |x - 4| \} \)
- \( \{ x \in \mathbb{R} | |x| > |x - 4| \} \)
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The Correct Option is B, C, D
Solution and Explanation
To determine which subsets of \( \mathbb{R} \) are connected, we need to understand the definition of a connected set. In the real numbers, a set is connected if it is an interval. Here's the step-by-step analysis of each subset in the options:
- Subset: \( \{ x \in \mathbb{R} \mid x^2 + x > 4 \} \)
- Start by solving the inequality: \( x^2 + x - 4 > 0 \).
- Factor the quadratic: \( (x - 2)(x + 2) > 0 \).
- Determine the intervals where this inequality holds: the solution is \( x \in (-\infty, -2) \cup (2, \infty) \).
- This is not a single interval; rather, it's a union of two disjoint intervals. Hence, this subset is not connected.
- Subset: \( \{ x \in \mathbb{R} \mid x^2 + x < 4 \} \)
- Start by solving the inequality: \( x^2 + x - 4 < 0 \).
- Factor the quadratic: \( (x - 2)(x + 2) < 0 \).
- Determine the intervals where this inequality holds: the solution is \( x \in (-2, 2) \).
- This is a single interval, hence this subset is connected.
- Subset: \( \{ x \in \mathbb{R} \mid |x| < |x - 4| \} \)
- Rewrite the inequality: \( x^2 < (x-4)^2 \).
- Expand and simplify: \( x^2 < x^2 - 8x + 16 \), which gives \( 8x < 16 \) or \( x < 2 \).
- The solution is \( x \in (-\infty, 2) \), which is a single interval. Thus, this subset is connected.
- Subset: \( \{ x \in \mathbb{R} \mid |x| > |x - 4| \} \)
- Rewrite the inequality: \( x^2 > (x-4)^2 \).
- Expand and simplify: \( x^2 > x^2 - 8x + 16 \), which gives \( 8x > 16 \) or \( x > 2 \).
- The solution is \( x \in (2, \infty) \), which is a single interval. Thus, this subset is connected.
In conclusion, the subsets that are connected are:
- \( \{ x \in \mathbb{R} \mid x^2 + x < 4 \} \)
- \( \{ x \in \mathbb{R} \mid |x| < |x - 4| \} \)
- \( \{ x \in \mathbb{R} \mid |x| > |x - 4| \} \)
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