Sets, Relations & Functions
Types of Sets
Grade 11
Question:
<p>Which of the following is a singleton set?</p><p>(a) \( \{ x : x < 1, x \in \mathbb{I} \} \)</p><p>(b) \( \{ x : x = 5, x \in \mathbb{I} \} \)</p><p>(c) \( \{ x : x = 1, x \in \mathbb{I} \} \)</p><p>(d) \( \{ x : x^2 + x + 1 = 0, x \in \mathbb{R} \} \)</p>
<p>(a) \( \{ x : x < 1, x \in \mathbb{I} \} \)</p>
<p>(b) \( \{ x : x = 5, x \in \mathbb{I} \} \)</p>
<p>(c) \( \{ x : x = 1, x \in \mathbb{I} \} \)</p>
<p>(d) \( \{ x : x^2 + x + 1 = 0, x \in \mathbb{R} \} \)</p>
Step-by-Step Solution
Key Concept: A singleton set contains exactly one element. Evaluate the set-builder notation to determine the elements in each set.
<p><strong>Step 1:</strong> Evaluate each option.</p><p>(a) $ \{ x : |x| < 1, x \in \mathbb{I} \} = \{ x : -1 < x < 1, x \in \mathbb{I} \} = \{0\} $ — This is a singleton set.</p><p>(b) $ \{ x : |x| = 5, x \in \mathbb{I} \} = \{ x : x = \pm\sqrt{5} \} $ — Since $\sqrt{5}$ is irrational, this set is empty.</p><p>(c) $ \{ x : x^2 = 1, x \in \mathbb{I} \} = \{ x : x = \pm 1\} = \{\pm 1\} $ — This has 2 elements.</p><p>(d) $ \{ x : x^2 + x + 1 = 0, x \in \mathbb{R} \} = \{ x : x = \frac{-1 \pm \sqrt{1-4}}{2}, x \in \mathbb{R} \} = \{ x : x = \frac{-1 \pm \sqrt{3}i}{2}, x \in \mathbb{R} \} = \emptyset $ — This is empty since the roots are complex.</p><p>∴ Answer is (a).</p>
Correct Answer: A