Question

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

• A. The set \(\{x \in \mathbb{R} : x \text{ is irrational}\}.\)
• B. The set \(\{x \in \mathbb{R} : x^3 - 1 \ge 0\}.\)
• C. The set \(\{x \in \mathbb{R} : x^3 + x + 1 \ge 0\}.\)
• D. The set \(\{x \in \mathbb{R} : x^3 - 2x + 1 \ge 0\}.\)

Hint:

In \(\mathbb{R}\), connectedness means being an interval (continuous block of points). Check the sign changes of polynomials to find such intervals.

Answer 1

The Correct Option is B, C

Step 1: Recall property.
A subset of \(\mathbb{R}\) is connected if and only if it is an interval (or a single point).
Step 2: Analyze each option.
(A) The irrationals are dense but not an interval — hence disconnected.
(B) \(x^3 - 1 \ge 0 \Rightarrow x \ge 1\), so the set \([1, \infty)\) is connected.
(C) \(x^3 + x + 1 \ge 0\) — since the cubic has exactly one real root, say \(\alpha\), the set is \([\alpha, \infty)\), which is connected.
(D) \(x^3 - 2x + 1 \ge 0\) has three real roots; hence the solution set is a union of disjoint intervals — disconnected.
Step 3: Conclusion.
Therefore, (B) and (C) are connected subsets.