Question

Which one of the following subsets of \(\mathbb{R}\) has a non-empty interior?

• A. The set of all irrational numbers in \(\mathbb{R}\).
• B. The set \(\{a \in \mathbb{R} : \sin(a) = 1\}\).
• C. The set \(\{b \in \mathbb{R} : x^2 + bx + 1 = 0 \text{ has distinct roots}\}\).
• D. The set of all rational numbers in \(\mathbb{R}\).

Hint:

A subset of \(\mathbb{R}\) has non-empty interior only if it contains an open interval.

Answer 1

The Correct Option is C

Step 1: Analyze each set.
(A) and (D): Both rationals and irrationals are dense but have empty interiors, since no interval in \(\mathbb{R}\) consists solely of rationals or irrationals.
(B) The set where \(\sin(a)=1\) is discrete (\(a = \pi/2 + 2n\pi\)), hence no interior.
(C) For \(x^2 + bx + 1 = 0\) to have distinct roots, discriminant \(b^2 - 4>0 \Rightarrow |b|>2.\) Thus, the set is \((-\infty, -2) \cup (2, \infty)\), which has open intervals ⇒ non-empty interior.
Step 2: Conclusion.
Hence, option (C) is correct.