Question

Which of the following subsets of \( \mathbb{R} \) is (are) connected?

• A. \( \{ x \in \mathbb{R} | x^2 + x > 4 \} \)
• B. \( \{ x \in \mathbb{R} | x^2 + x < 4 \} \)
• C. \( \{ x \in \mathbb{R} | |x| < |x - 4| \} \)
• D. \( \{ x \in \mathbb{R} | |x| > |x - 4| \} \)

Hint:

For connected sets, look for intervals or regions without breaks or gaps. Disjoint sets are not connected.

Answer 1

The Correct Option is B, C, D

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:

1. 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.
2. 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.
3. 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.
4. 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:

• \( \{ x \in \mathbb{R} \mid x^2 + x < 4 \} \)
• \( \{ x \in \mathbb{R} \mid |x| < |x - 4| \} \)
• \( \{ x \in \mathbb{R} \mid |x| > |x - 4| \} \)